Handbook of Geometric Topology

Piecewise linear topology

247

B.

Fig. 3.

Finally, we extend the idea of a handle decomposition to a cobordism between (« — 1)manifolds. A cobordism is a triple {W\ M, M'), where W is a PL ^-manifold and dW = M VJ M\ M D M^ = 0. A handle decomposition of W, rel M, is a presentation W = CU HQU HiU '"U HnUC\ where C and C are regular neighborhoods of M and M in W, respectively, and Wp C U U/^/7 ^i is obtained from iy^_i by adding /7-handles. Dual handle decompositions. Notice that if W = CU HQU H\ U-- -U HnUC^ isa. handle decomposition of a cobordism (W; M, MO, and if H^P^ is a /7-handle with characteristic map /i, then H^P^ H (U/>p /// U C) = h(JP x dT-P). That is, H^^^ can be thought of as an (n - /7)-handle added to |J,^^ Hi U C\ With this point of view, we write H^P^ = f}(n-p) ^^^ ^^11 ^(«-p) the dual (n - p)-handle determined by H^P\ Thus, we also get a handle decomposition W = C'U HQU HiU---U HnUC, where Hp = Hn-p. Dual handle structures are closely related to dual cell structures described in Section 2. Handle decompositions arising from triangulations of a manifold are generally too large to be of much use, although they often provide a place to get started. The goal is to try to find the simplest possible handle decomposition. For example, if a cobordism {W\ M, MO has a handle decomposition with no handles, then \y = C U CMs a product: {W\ M, MO = (M X /; M X {0}, M X {1}). An obvious necessary condition for this to happen is that the inclusions M/ ^-^ W, / = 0,1, are homotopy equivalences. A cobordism {W\ M, M') satisfying this condition is called an h-cobordism between M and M', or simply an /z-cobordism. /z-COBORDISM THEOREM 6.1. Suppose {W\ M, MO is a compact h-cobordism, dim W ^ 6, and W is simply connected. Then {W\ M, MO = {M x I\ M x {0}, M x {1}). We shall outhne a proof of the /z-Cobordism Theorem in this section. Our treatment is taken from [50], where many of the omitted details may be found.

248

J.L. Bryant

Simplifying handle decompositions. For the immediate discussion, we will let (W; M, M0 denote a compact cobordism with dim W = n. Our first observation is that "sliding a handle" does not change the topology of the resulting manifold. 6.2. If f,g:dIP x /"~^ -^ dM' are ambient isotopic attaching maps, then W \Jf H^P^ = W yjg H^P^ by a homeomorphism that is fixed outside a regular neighborhood (collar) ofM'. LEMMA

By Proposition 3.12, an isotopy of M' extends to W, fixing the complement of a collar on M^ D PROOF.

LEMMA

6.3. Ifp^q,

then (W U H^^'>) U H^P'> = (WUH^P'>) U H^'^\ with H^P"^ and i/^^>

disjoint. PROOF. Let Sa be the attaching sphere for H^P'> and Sb the belt sphere for H^^\ Then dim5^ + dim5^ = (p — I) -\- (n — q — I) < n — I so that Sa can be general positioned to miss Sb in d(W U //^^^). Use 3.14 and Lemma 6.2 to "squeeze" the p-handle so that H^P^ nSb = 0 as well. Let A^ be a regular neighborhood of the cocore of H^^^ in W U H^^^ such that N n H^P"^ = 0. Since H^^^ is also a regular neighborhood, there is an isotopy of W U if (^^ taking A^ to /f (^\ This isotopy slides 77^^^ off of / / ( ^ \ so Lemma 6.2 applies to complete the proof. D

As a consequence of Lemma 6.3, we can rearrange the addition of handles to a cobordism W so that the handles are added in nondecreasing order, thereby producing a handle decomposition of W. We now look at circumstances in which handles may be eliminated. Suppose Wi=WUH^P'>[JH^P^^\dimW = n. Then //^^^ and //(z'+i) are called complementary handles if the attaching sphere Sa of H^P^^^ meets the belt sphere Sb of H^P^ transversely in a single point. This means that near x = SaH Sb, and after an ambient isotopy of the attaching map / for H^P~^^\ f matches up the product structure on dH^P^^"^ with that on a//^^>. LEMMA 6.4. Suppose that W\=W^ H^P^ U H^P^^\ where H^P^ and H^P^^'> are complementary handles. Then W\ =W by a PL homeomorphism that is the identity outside a collar on M'. PROOF. Let h:JPx

r~P

-> H^P^ be the characteristic map for H^P'> and let

/ : (djP^^) X r-P-^ -^ d{W U H^P^) be the attaching map for H^P^^\ Lemma 6.2, we may assume that h{jp X ({1} X r-p-^))

Using the Regular Neighborhood Theorem and

= f{{jp X {1}) X

r-p-^),

Piecewise linear topology

\

\

\

K

^//' ^

1 V

—

"""''

249

r

1^-.

(P+n

h{j\{[\]xr''''))

1

/((/x{l))x/"^"' )

L __.

X

Fig. 4.

and that f~^ o h\JP x ({1} x J^'-P'^) = id. (See Figure 4.) Thus, we see that Wi is obtained from W by attaching an n-ball B = H^P^ U //(Z^+D to M' along an (n - l)-ball in a^. That is, Wi shells to W. The reverse of this process allows one to introduce a cancelling pair of handles to a cobordism. In general, if Wi = W U H^P^ U H^P^^\ then we can define the incidence number £{H^P^^\ H^P"^) as follows. There is a strong deformation retraction of \y U H^P"^ onto W^h{IP X {0}), where h is the characteristic map for H^P\ The composition WU H^P^ -^ W U h{IP X {0}) ^ {Wyj h(IP X {0}))/W = SP gives a mapping g : W U H^P^ -^ SP. If 5^ is the attaching sphere for H^P^^\ then the restriction g\Sa gives a mapping of 5^ = 5« -> SP. Choose orientations for (each) 5^, and define 6(H^P^^\ H^P'>) to be the degree of this map. Thus, £{H^P^^\ H^P"^) is an integer, which is well-defined up to sign. If Sa and the belt sphere Sb for H^P^ are in general position in d{W U H^P^), then they intersect transversely in a finite number of points. If H^P"^ and H^P~^^^ are given orientations, then e(H^P~^^\ H^P^) is the algebraic intersection number of Sa and Sb in d{W U //(^>). If H^^) and H^P+^'> are a complementary pair of handles, then, clearly, e{H^P^^\ H^P^) = ±1. The next lemma gives conditions under which, up to an ambient isotopy of attaching maps, the converse is true. The proof uses another form of the Whitney Trick and may be found in [50, Chapter 6]. D LEMMA 6.5 (Handle Cancellation Lemma). Suppose Wi = (W U H^P^) U /

H^P-^^\

2^p^n-4, n ^ 6, M' is simply connected, and S(H^P^^\ H^P^) = ± 1 . Then the attaching map f for H^P^^^ is ambient isotopic to an attaching map g such that, in W2 = {Wyj H^P^) Ug H'^P+^\ H^P"^ and H^P^^^ are complementary handles. Thus, Wi=W2 = W. (p) rip) LEMMA 6.6 (Handle Addition Lemma). Suppose W\ =W U/^ 7/|^^ U/2 H\^\ where M. (~^ H2 =&, l^p^n H^ — 2, and M' is simply connected. Then f\ is ambient isotopic to /3, where [/s] = [/i] + [fi\ in 7Tp{M').

250

J.L. Bryant

\i2^p^n — 2 and M' is connected, we can connect PL embedded {p — 1)spheres S\ and ^2 with a PL "ribbon" D = g(I x /^~^) in M\ where ^ is a PL embedding, g(I X IP-^)nSi = g({0} X IP-^) mdg(I X IP-^)nS2 = gi{l} X IP-^).iSec[50,Ch2iVter 5].) In this way we can add the homotopy classes of [f\] and [/2] in np-\(M^) (this requires 2 ^ p ^ n — 2), and if M' is simply connected, the resulting class is independent of D. Inside the boundary of the /^-handle H\ = H\ in W\, there is a "parallel" copy of its core: B = h(JP x {x}), where x e dJ^~P and his a. characteristic map for H\. Connect the boundary sphere 5 of 5 to the attaching sphere 52 for the handle H2 with a ribbon D in M^ Now we have the collapses 5 U D U ^2 \ ^2 and 5 U D U ^2 \ ^3 = (^i U D U ^2) g(/ xint/^~^) inM2 = a ( W U / / i ) - M . Thus, in M2, the Regular NeighborhoodTheorem provides an ambient isotopy of the attaching map f\ to f^ : SP~^ -^ M' with [/a] = [/i] + [/2] in 7tp {M'). The new handle can be moved off of H[^^ U H^f"*. D PROOF.

LEMMA 6.7. Suppose that {W\ M, M^) is a connected cobordism. Then W has a handle decomposition with no 0- or n-handles. PROOF. If a 1-handle H^^^ joins C to a 0-handle H^^\ then (H^^\ H^^^) is a complementary pair. Proceed by induction: if there are any 0-handles, then there is one that is joined to C by a 1-handle. The n-handles in a handle decomposition of W are the 0-handles in the dual decomposition. D

If M is simply connected, one can use a handle decomposition (W; M, M^) = CU H\U • • • U H(n-\) U C\ where Hp = H^^^ U • • • U //^^^ is the (disjoint) union of handles of index /?, to compute the homology of the pair (W; M). Form the chain complex whose /7th chain group, Cp, is generated by the (oriented) /7-handles. If //-^ is a p-handle, define a(///^^) = J2j £(H^^\ Hf~^^)Hf~^\ Observe that if X D F is a polyhedron with dim(X — F) ^ /?, and if / : (X, Y) -^ {W, M) is a mapping, then / is homotopic, rel F, to a mapping g'.X -> CVJ H\\J - "\J Hp. Just proceed inductively: use general position to get f{X) disjoint from the cocores of higher-dimensional handles and then use the handle structure to get f{X) miss the handles themselves. A similar general position argument can also be used to prove the following useful fact. 6.8. Suppose that W = CUHoU' • -UHn^C^ isa handle decomposition, W^^^ = C[JHiUHiU"-UHp,and M^P^ = dW'^P^ - M. Then (a) 7Ti(W, W^P^) = Ofor i < p, and (b) m (W, M^P^) = Ofor i ^ min{/7, n-p-\}.

LEMMA

6.9. Suppose that W is connected, n ^ 6, M is simply connected, and (W; M, M') has a handle decomposition with no handles of index < p, I ^ p ^ n — 4. If Hp{W, M) = 0, then there is another handle decomposition with no handles of index ^ p and with the same number of handles of index > p -\-1. LEMMA

PROOF. Let W = C U //i U • • • U //^ U C be a handle decomposition for W. Case 1: /7 = 1. Let H^^^ be a 1-handle in H\; //^'^ is attached to Mo = dC - M. Let a be an arc in 9//^^^ parallel to its core, and let ^6 be a PL arc in C joining the endpoints of of.

Piecewise linear topology

251

Use general position to get a to miss all 2-handles and p to miss the cores of the 1- and 2-handles so that y =aU fi c. d(C U H^^^) — M. By Lemma 6.8 and general position, y bounds a PL disk D in Mi = 3 (C U //i U i/2) - M. By 4.6 D is locally flat in M\. Introduce a complementary pair of 2- and 3-handles H^^^ and H^^^ so that H^^^ has attaching sphere 9D and D lies in the attaching sphere of H^^\ Then //^^^ and //^^^ are complementary handles and may be eliminated. After rearranging handles, we get a new decomposition with //^^^ eliminated and a new H^^'^ introduced. Case 2: 1 < p < « — 4. As Hp{W, M) = 0 and there are no handles of index < /?, the boundary map 9 : Cp+i -> C^ must be onto. If H^P^ is a p-handle, then

for some chainj^-w/i//^"^^^ in Cp+i. Thus we can use handle addition to get e{H^ ^ H^P)) = 1 for some (p + l)-handle i//^+^^ and s(H^/^^\ then cancel H^P'> and ///^+^\

H^P"^)

= 0 for j ^ i. We may D

PROOF OF THE /Z-COBORDISM THEOREM. Let {W\ M, M') be a compact /z-cobordism, dim W ^ 6, with W is simply connected. Let (1^; M, MO = C U //i U • • • U //«_i U C be a handle decomposition with no 0- or w-handles. Use Lemma 6.9 to eliminate handles of index < « — 4, leaving only handles of index (/i — 3), (w — 2), and {n — \). Thus the dual handle decomposition only has handles of index 1, 2, and 3. Eliminate the dual 1handles from the dual decomposition, leaving only dual 2- and 3-handles. Since we are working over the integers, the matrix of the boundary map from C3 to C2, which is invertible over Z, may be diagonalized, and the elementary operations may be realized by handle additions. Hence, we arrive at a handle decomposition with only complementary handles in dimensions 2 and 3, which may be cancelled as well, leaving a decomposition with no handles. Hence, W is a product M x / . D COROLLARY 6.10 (Strong Poincare Conjecture [3,55]). 7/*M is a k-connected, closed PL n-manifold, n^6, k= In/l}, then M is PL homeomorphic to 5". PROOF. Remove the interiors of disjoint, PL n-balls from M. The result is an /i-cobordism between boundary spheres, which is a product 5"~^ x / . D

If W is not simply connected, one must define incidence numbers as elements of the group ring Z[7ri (M)], and the proof of the /i-cobordism theorem may break down at the last step, when we eliminate the dual 2- and 3-handles. In this case the matrix of the boundary map is a non-singular matrix over Z[n\(M)], and it can be diagonalized if, and only if, we are given the additional hypothesis that the inclusion M c W is a simple homotopy equivalence. An /i-cobordism (W; M, MO with this property is called an s-cobordism. 6.11 (5-Cobordism Theorem). Suppose that {W\ M, M') is an h-cobordism, n^6. Then (W; M, MO = (MxI;Mx {0}, M x {1}) if, and only if (W; M, M') is also an s-cobordism. THEOREM

252

J.L. Bryant

There are also relative versions of the h- and ^-cobordism theorems. Their proofs proceed very much the same as in the absolute case. A relative cobordism between {n — \)manifolds M and M' with boundaries is a triple {W\ M, M'), where W is an n-manifold such that aW = M U y U M^ where M fi M' = 0, a v = aM U dM\ V iM^ = dM, and V n M^ = aM^ A relative cobordism is an /i-cobordism (5-cobordism) if all of the inclusions M,M' c. W, dM,dM^ c. V are homotopy equivalences (simple homotopy equivalences). THEOREM 6.12 (Relative ^-Cobordism Theorem). Suppose W is a compact relative hcobordism, « ^ 6, such that V~dMxL Then (W; M, M') = (MxI;Mx {0}, M x {1}) if, and only if, (W; M, M') is also an s-cobordism.

7. Isotopies, unknotting In this section we explore further the question: Given polyhedra X and F, when are homotopic embeddings f,g:X-> Y ambient isotopic? The first result is Zeeman's unknotting theorem [65]. THEOREM 7.1 ([65]). A PL sphere pair (S"", 5^) or a proper PL ball pair {B^'.B'i), n q ^3 is unknotted. PROOF ([50]). The proof is by induction (eventually) on n. The case n = 3 is trivial. The case n = 4 follows from Theorem 4.5(i), for sphere pairs, and Theorem 4.7, for ball pairs. Theorem 4.5(ii) gives the result for sphere pairs when n = 5, and the Regular Neighborhood Theorem for Pairs can, in turn, be used to show that ball pairs unknot. Assume, then, that n ^ 6. Let (B"", 5^) be a ball pair, n-q ^3. By induction, ( 5 " , 5^) is locally flat. Hence if A^ is a regular neighborhood of B^ in 5", then, by 3.22, (A^, B^) is an unknotted ball pair. Let W = Ci(B'' -N),M = NnW. Then W is a relative cobordism between (M, SM) and (M^ SM'), where M' = Cl((dB'' nW)N') and A^' is a small collar of dM in dB^ Pi W. Since B^ is contractible and B^ — B^ is simply connected, by general position, W is an /i-cobordism. Thus W = M x I and so ( 5 " , B^) is unknotted. Suppose (S^, 5^) is a sphere pair, n ^ 6, and n — q ^3. Let A' > L be a triangulation of S" D S^, and let f be a vertex of L. Then (st(i;, K), si(v, L)) is an unknotted ball pair, as is the complementary pair. Thus (S^, 5*^) is obtained by gluing two unknotted ball pairs together. • COROLLARY

7.2. Any proper (n, q)-manifold pair is locally fiat, ifn — q ^3.

Since the Whitehead group of Z is trivial, any /i-cobordism between manifolds with fundamental group Z is an 5-cobordism. Thus the proof of Theorem 7.1 works for locally flat sphere pairs in codimension 2, provided the pair is "homotopically unknotted". 7.3. Suppose (5"", 5"""^), n ^ 6, is a locally flat sphere pain Then (S'',S''~^) is unknotted if, and only if 5"" — 5''^"^ has the homotopy type of a circle.

THEOREM

Piecewise linear topology

253

We consider next the weaker question: When are isotopic embeddings of X into M ambient isotopic? (See Section 3 for definitions.) The answer is yes whenever the isotopy is "locally extendable". Given an isotopy F : X x / ^- M x / (or a proper isotopy of pairs) and a point (jc, 0 ^ ^ x / . ^ is locally extendable at (x, t) if there are neighborhoods V of jc in X, U of F(jc, t) in M, and J of tin I and a level preserving embedding h:U x J ^^ M X J such that h(y, t) = (y, t) for 2i\\ y eU and h(F(x, s), s) = F(x, s) for (jc, s) G y X / . In other words, local collars of F(X x {t}) in F(X x [t, 1]) (if t < 1) and in F(X X [0, t]) (if t > 0) are extendable (locally) to collars of M x {t} in M x [t, 1] and M X [0, t]. By Theorem 3.19, a locally extendable isotopy is extendable at X x [t] for all / e /, meaning we can choose V = X and U = M. This fact together with a standard compactness argument proves the following extension theorem. 7.4. If X is a compact polyhedron and F.X -^ Y is a locally extendable isotopy, then F is ambient.

THEOREM

If Q and M are PL q- and n-manifolds, respectively, with n—q^3, then, by Zeeman's Unknotting Theorem, any isotopy of QinM is locally extendable. This fact, together with Proposition 3.12 establishes the following corollary. COROLLARY 7.5 ([32]). Suppose Q and M are PLq- and n-manifolds, respectively, with n — q ^3, and Q is compact. Then any proper isotopy of(Q,dQ) in (M, dM) is ambient.

If X and Y are polyhedra and X is compact, a proper PL embedding / : ((f * X), X) ^(w ^Y,Y) is unknotted if there is a PL homeomorphism / i : u ; * y ^ - w ; * 7 fixing Y such that hf is the cone on f\X. Lickorish's Cone Unknotting Theorem [38] is a polyhedral analogue of Theorem 7.1. We state it next without proof. 7.6 ([38]). Suppose that X is a compact (q — I)-dimensionalpolyhedron and / : (f * X, X) -> (B^, S^~^) is a proper embedding, n — q ^3. Then f is unknotted. THEOREM

Theorem 7.6 allows one to prove that a PL isotopy of a polyhedron in a PL manifold in codimension ^ 3 is locally extendable. Thus, the line of argument above can be used to prove the following isotopy extension theorem due to Hudson [30]. COROLLARY 7.7 ([30]). Suppose that X is a compact q-dimensionalpolyhedron, M is a PL n-manifold, n — q^3. Then any isotopy ofX in M is ambient.

An embedding F.XxI^-YxI satisfying F'^Y x {/}) = X x {/}, / = 0,1, is a concordance of X in y (from FQ to F \ ) . Thus, an isotopy of X in 7 is a level-preserving concordance. Hudson [31] obtains the following improvement of Corollary 7.7. Rourke [47] gives a "handle straightening" argument for this result as well. THEOREM 7.8 ([31]). Suppose that X is a compact q-dimensionalpolyhedron, M is a PL n-manifold, n — q ^3. Then concordant embeddings ofX in M are ambient isotopic.

254

J.L. Bryant

COROLLARY 7.9. Suppose that Q is a compact PL q-manifold, M is a PL n-manifold, n — q ^3, and f :(Q,dQ) -^ (M, dM) is a (proper) PL embedding such that f is (Iq — n + I)-connected. If g :(Q,dQ) ^^ (M, dM) is a PL embedding that is homotopic to f rel 9 Q, then f and g are ambient isotopic.

8. Approximations, controlled isotopies Many of the results of Sections 5, 6, and 7 have "controlled" analogues. In this section we state without proof a few of the basic theorems of this type. The first result is Miller's Approximation Theorem [42]. 8.1 ([42]). Suppose that Q is a PL q-manifold, M is a PL n-manifold, n — q ^3, and f :iQ,dQ) -^ (M, SM) is a topological embedding. Then for every e.Q^-iO, oo), there is a PL embedding g'.(Qy^Q)-^ (^, 9M) such that d(f, g) < e. THEOREM

(This result was initially announced by Homma [27], but a problem was discovered in his proof by H. Berkowitz. A proof along the lines originally presented by Homma can be found in [9].) Using Miller's theorem for ^-cells, Bryant [8] was able to extend Miller's theorem to embeddings of polyhedra. 8.2 ([8]). Suppose that X is a q-dimensionalpolyhedron and f :X -^ M is a topological embedding into a PL n-manifold. Then for every s: Q ^ (0, oo), there is a PL embedding g:X-^M such that d(f, g) < s. THEOREM

The next theorem is an e version of the isotopy theorems of Section 7. This result is an amalgamation of results due primarily to Connelly [18] and Miller [41], with contributions and improvements due to Cobb [14], Akin [1], and Bryant and Seebeck [10]. 8.3 ([18,41]). Suppose that (X, Y) is a polyhedral pair, dimF < dimX = q, M is a PL n-manifold, and f:{X,Y) -^ {M,dM) is a proper topological embedding, n — q ^3. Then for every e:X ^ (0, oo) there is a 8:X -> (0, oo) such that if gi: (X, Y) —> (M, dM) are PL embeddings, i = 0,1, within 8 of f, then there is an s-push H of M such that Hgo = g\. THEOREM

There are useful variations on Theorem 8.3. For example, if dim(X — Y)^n — 3,f\Y is a PL embedding, and gi\Y = f\Y, i = 0,1, then one can get an e-push H of M rel dM wiihHgo = g\. There are a number of controlled versions of the engulfing theorem (Theorem 5.2), although mostly they have been replaced by Quinn's End Theorem [46]. Bing's article "Radial Engulfing" [5], contains a variety of such theorems the reader is encouraged to survey. We state one such result from [10]. It requires a definition: A subset F of a space X is 1-LCC (1-locally co-connected) in X if for each x sY and each neighborhood U of x in X, there is a neighborhood V of jc in X such that the inclusion 7t\ (V — Y) ^^ 7i\ (U — Y) is trivial. An embedding f:Y->X is 1-LCC if / ( F ) is 1-LCC in X.

Piecewise linear topology

255

THEOREM 8.4 ([10]). Suppose fiX^Misa 1-LCC embedding of a q-dimensional polyhedron X into a PL n-manifold M, n — q ^3, n^ 5. Then for every 6 > 0, there is a 5 > 0 such that if g'.X ^^ M is aPL embedding within S of f and U is any neighborhood of g{X), then there is an s-push H of M such that H\(f(X)) C U.

If X is not compact then s and 8 are functions of X. The 1-LCC condition allows one to push fiX) off of the 2-skeleton of a neighborhood of f(X) and then close to the dual (n — 3)-skeleton. One then engulfs the dual (n — 3)-skeleton with U. These are e versions of StaUings arguments in [57]. Using Theorems 8.2, 8.3, and 8.4, and an infinite process due to Homma [26] and Gluck [23], one can deduce the following "taming" theorem of Bryant and Seebeck [10]. THEOREM 8.5 ([10]). Suppose f:X-^Misa 1-LCC embedding of a q-dimensional polyhedron X into a PL n-manifold M, n —q ^3, n^5. Then for every e > 0, there is an s-push H of M such that H\ f is PL.

There is a 4-dimensional analogue of this result due to R.D. Edwards (unpublished), obtained from Casson-Freedman handle theory for 4-manifolds [21].

9. Triangulations of manifolds We conclude this chapter with a discussion of the two most central issues in PL topology: existence and uniqueness of triangulations of topological manifolds. Classically, these questions dealt with PL triangulations of manifolds, although there are obvious related questions concerning triangulations in general. Existence and uniqueness of PL triangulations of manifolds of dimensions ^ 3 have been known for some time, the case n = 3 being due to Moise [44] and Bing [4]. The uniqueness question (the Hauptvermutung) can be asked for polyhedra in general. Milnor was the first to construct examples of polyhedra that are homeomorphic, but not piecewise linearly homeomorphic [43]. Edwards was able to exhibit a non-combinatorial triangulation of 5", n ^ 5, by showing that the A:-fold suspension of a non-simply connected 3-manifold constructed by Mazur [40] is topologically homeomorphic to S^^^. (See [19].) Edwards and Cannon [11] solved the "Double Suspension Problem" in general by proving the following theorem. THEOREM 9.1 ([11]). Suppose that H^ is a PL n-manifold with the homology ofS^. Then, for each k^ I, the polyhedron 5^ * / / is topologically homeomorphic to 5^+^^+!.

Whenever H is not simply connected, which can happen when « ^ 3, the polyhedron S^^H is not PL homeomorphic to the standard 5^+"+i. Thus, uniqueness of triangulations of topological manifolds fails if one does not require triangulations to be PL. The problem of uniqueness of PL triangulations is more subtle yet. Suppose that M and A^ are PL n-manifolds and h'.M^^Nisa. topological homeomorphism. Sullivan's idea [58,59] was to prove that M and N are PL homeomorphic by taking a handle decomposition of M and, inductively, "straightening" their images under h. This

256

J.L. Bryant

idea presents a handle problem, that is, a topological homeomorphism h:B^ x M"~^ -^ V" onto a PL manifold V^ that is PL on a neighborhood of S^~^ x R"~^. The handle can be straightened if there is an isotopy / / of V", fixed on a neighborhood of S^~^ x R"~^ and outside a compact set, such that H\h is PL on /z: 5^ x B"~^. Sullivan showed that, for n ^ 5, there was a possible Z/2 obstruction to straightening 3-handles [58]. In his solution to the annulus conjecture, Kirby [36] showed how to straighten 0-handles when n ^ 5. Kirby and Siebenmann (see [37]) proved that /:-handles can be straightened provided n ^ 5 and k ^3. Whether or not 3-handles could be straightened depended upon the Hauptvermutung for the n-torus T^ = S^ x - • • x S^. The following result of Hsiang and Shaneson [28], Wall [60] and Casson classifies the PL structures on T^, showing, in particular, that they are not all equivalent. 9.2. For n^ 5, the set of PL equivalence classes of PL manifolds topologically homeomorphic to T^, is in one-to-one correspondence with the set of orbits of(A^~^Z^)(S> Z/2 under the natural action of GL{n, Z). The standard torus corresponds to the zero element under this action.

THEOREM

{A^V denotes the k\h exterior algebra on Z".) In particular there are non-standard T^ 's. The classification implies that if M is a non-standard, or fake, torus, then even covers of M will be standard, while odd covers are not. Kirby [36] used the first fact in his "torus trick" to prove the annulus conjecture, while the second fact is used to disprove the Hauptvermutung. (See [37].) 9.3 ([37]). Given a PL n-manifold M, n^ 6 or n^ 5 if dM = 0, the isotopy classes of PL structures on M are in one-to-one correspondence with the elements of //3(M;Z/2). THEOREM

The coefficient group Z/2 appears as the homotopy group TTB(TOP/PL), where TOP/PL is the fiber of the forgetful map ^TOP -> ^PL of classifying spaces for topological and piecewise linear bundles. Further handle analysis leads to the following existence theorem of Kirby-Siebenmann. 9.4 ([37]). Given a topological n-manifold M,n^6,orn^5ifdM has a PL triangulation, there is a well-defined obstruction in H^{M\ Z/2) to triangulating M as a PL manifold extending the triangulation on dM. THEOREM

There are, in fact, topological 4-manifolds that do not admit PL triangulations. One such may be constructed using results of Kervaire and Freedman. Kervaire shows [35] that there is a homology 3-sphere / / , the Poincare 3-sphere, that bounds a parallelizable PL 4-manifold M with signature 8. Freedman shows [21] that H bounds a contractible topological 4-manifold M'. V = M U// M' is then a closed 4-manifold with signature 8. A result of Rohlin (see [34]) states that V cannot have a PL triangulation, for, if it did, its signature would be divisible by 16. (A manifold is parallelizable if its tangent bundle is a product bundle.)

Piecewise linear topology

257

It is clear that if M has a PL triangulation, then so does M x R^ for ^ > 1. KirbySiebenmann prove a very strong converse to this fact for higher dimensional topological manifolds. 9.5 (Product Structure Theorem [37]). Suppose that M is a topological n-manifold and that M xR^, k^ I, is triangulated as a PL manifold. If n ^ 6, or n ^ 5 and 9M = 0, then there is a PL triangulation of M inducing an equivalent triangulation onM X R^ THEOREM

There are relative versions of the Product Structure Theorem, which the reader may find in [37, I, Section 5]. The following, rather obvious corollary has proved useful in applications. COROLLARY 9.6. Suppose that M is a topological n-manifold with boundary, n^6, that int M has a PL triangulation. Then M has a PL triangulation.

such

In particular, any topological, n-dimensional submanifold (with boundary) of a PL nmanifold M, n ^ 6, has a PL structure. Likewise, if a compact subset C of a topological n-manifold, « ^ 6, has vanishing Cech cohomology in dimension 4, then naturality of the obstruction in Theorem 9.4 implies that sufficiently small manifold neighborhoods of C have PL structures. The question as to whether a topological n-manifold has a triangulation (PL or not) has been investigated extensively by Galewski and Stem. (See, e.g., [22].) Let 0^ denote the group obtained from oriented PL homology 3-spheres, under the operation of connected sum, #, modulo those that bound acyclic PL 4-manifolds. There is a homomorphism fi-.O^ ^ Z/2 (the Kervaire-Milnor-Rohlin map) defined by /x(^) = [a(W)/Sl where a(W) denotes the signature of any parallelizable 4-manifold W with boundary H. If H is the Poincare homology 3-sphere, then /x(/i/) = 1, so that /x is surjective. THEOREM 9.7 ([22]). Suppose M is a topological n-manifold, n ^ 6 or n ^ 5 if dM is triangulated. Then there is an element IM ^ H^(M, dM; ker/x) such that IM =0 iff^there is a triangulation of M compatible with the given triangulation on dM. Moreover, the set of concordance classes of triangulations of M rel dM is in one-to-one correspondence with the elements of H^{M, dM; ker/x).

Triangulations KQ and Ki of M are concordant if there is a triangulation K of M x I restricting to Ki on Af x {/},/= 0,1. Galewski and Stem [22] and Matumoto [39] have shown that all compact topological «-manifolds (n > 6 or « ^ 5 if 9M is triangulated) can be triangulated if there is a homology 3-sphere H such that /x(//) = 1 and H#H bounds a parallelizable 4-manifold with signature 0. At the time of this writing it is unknown whether such a 3-manifold exists or whether every topological n-manifold, n ^ 5, can be triangulated. Casson, however, has found a topological 4-manifold that cannot be triangulated [13].

258

J.L. Bryant

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

E. Akin, Manifold phenomena in the theory ofpolyhedra. Trans. Amer. Math. Soc. 143 (1969), 413^73. P. Alexandroff and H. Hopf, Topologie, I, Springer, Beriin (1945). D. Barden, The structure of manifolds. Doctoral thesis, Cambridge Univ. (1963). R.H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37-65. R.H. Bing, Radial engulfing, Conf. on the Topology of Manifolds, J.G. Hocking, ed., Prindle, Weber, & Schmidt, Boston (1968). R.H. Bing and J.M. Kister, Taming complexes in hyperplanes, Duke Math. J. 31 (1964), 491-511. M. Brown, Locally flat embeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331-341. J.L. Bryant, Approximating embeddings ofpolyhedra in codimension three. Trans. Amer. Math. Soc. 170 (1972), 85-95. J.L. Bryant, Triangulation and general position of PL diagrams. Topology Appl. 34 (1990), 211-233. J.L. Bryant and C.L. Seebeck, Locally nice embeddings in codimension three. Quart. J. Math. Oxford (2) 21 (1970), 265-272. J.W. Cannon, Shrinking cell-like decompositions of manifolds in codimension three, Ann. of Math. (2) 110 (1979), 83-112. S. Cappell and J. Shaneson, Piecewise linear embeddings and their singularities, Ann. of Math. (2) 103 (1976), 163-228. A. Casson, unpubhshed work. J.L Cobb, Taming almost PL embeddings in codimensions 3, Abstract 68T-241, Notices Amer. Math. Soc. 15 (2) (1968), 371. M.M. Cohen, Simplicial structures and transverse cellularity, Ann. of Math. 85 (1967), 218-245. M.M. Cohen, A general theory of relative regular neighborhoods. Trans. Amer. Math. Soc. 136 (1969), 189-229. M.M. Cohen, A Course in Simple Homotopy Theory, Springer, New York (1970). R. Connelly, Unknotting close polyhedra in codimension three. Topology of Manifolds, J.C. Cantrell, ed., Markham, Chicago (1970). R.J. Daverman, Decompositions of Manifolds, Academic Press, Orlando (1986). A.I. Flores, tjber n-dimensionale Komplexe die im E^"+^ Absolut selbstverschlungen sind, Ergib. Math. KoUoq. 6 (1935), 4-7. M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357^53. D. Galewski and R. Stem, A universal 5-manifold with respect to simplicial triangulations. Geometric Topology (Proc. Georgia Topology Conf., Athens, GA, 1977), Academic Press, New York (1979), 345350. H. Gluck, Embeddings in the trivial range, Ann. of Math. (2) 81 (1965), 195-210. A. Haefliger, Knotted spheres and related geometric problems, Proc. I.C.M. (Moscow, 1966), Mir (1968), 437^U4. M.W. Hirsch, On tubular neighbourhoods of manifolds, Proc. Cambridge Philos. Soc. 62 (1966), 177-183. T. Homma, On the imbedding ofpolyhedra in manifolds, Yokohama Math. J. 10 (1962), 5-10. T. Homma, A theorem of piecewise linear approximations, Yokohama Math. J. 14 (1966), 47-54. W.-C. Hsiang and J. Shaneson, Fake tori. Topology of Manifolds, J.C. Cantrell, ed., Markham, Chicago (1970), 18-51. J.F.P Hudson, P I embeddings, Ann. of Math. (2) 85 (1967), 1-31. J.F.P. Hudson, Piece Linear Topology, W.A. Benjamin, New York (1969). J.F.P Hudson, Concordance, isotopy and dijfeotopy, Ann. of Math. (2) 91 (1970), 425^U8. J.F.P Hudson and E.G. Zeeman, On combinatorial isotopy, Publ. I.H.E.S. (Paris) 19 (1964), 69-94. M.C. Irwin, Embeddings ofpolyhedral manifolds, Ann. of Math. (2) 82 (1965), 1-14. M.A. Kervaire, On the Pontryagin classes of certain SO{n)-bundles over manifolds, Amer. J. Math. 80 (1958), 632-638. M.A. Kervaire, Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 (1969), 67-72. R.C. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. (2) 89 (1969), 575-582.

Piecewise linear topology

259

[37] R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Princeton Univ. Press, Princeton, NJ (1977). [38] W.B.R. Lickorish, The piecewise linear unknotting of cones. Topology 4 (1965), 67-91. [39] T. Matumoto, Varietes simpliciales d'homologie et varietes topologiques metrisables. Thesis, Univ. de Paris-Sud, Orsay (1976). [40] B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. (2) 73 (1961), 221-228. [41] R.T. Miller, Close isotopies on piecewise-linear manifolds. Trans. Amer. Math. Soc. 151 (1970), 597-628. [42] R.T. Miller, Approximating codimension 3 embeddings, Ann. of Math. (2) 95 (1972), 406-416. [43] J. Milnor, Two complexes that are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575-590. [44] E.E. Moise, Affine structures in 3-manifolds, V: The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96-114. [45] J.R. Munkres, Topology A First Course, Prentice-Hall, Englewood Cliffs, NJ (1975). [46] F. Quinn, Ends of Maps, I, Ann. of Math. (2) 110 (1979), 275-331. [47] C.P Rourke, Embedded handle theory, concordance and isotopy. Topology of Manifolds, J.C. Cantrell, ed., Markham, Chicago (1970). [48] C.P. Rourke and B.J. Sanderson, An embedding without a normal microbundle. Invent. Math. 3 (1967), 293-299. [49] C.P Rourke and B.J. Sanderson, Block bundles: I, Ann. of Math. (2) 87 (1968), 1-28. [50] C.P. Rourke and B.J. Sanderson, Introduction to Piecewise-Linear Topology, Springer, Berlin (1972). [51] A. Scott, Infinite regular neighborhoods, J. London Math. Soc. 42 (1963), 245-253. [52] A. Shapiro, Obstructions to the imbedding of a complex in euclidean space, Ann. of Math. (2) 66 (1955), 256-269. [53] L.C. Siebenmann, Infinite simple homotopy types, Indag. Math. 32 (5) (1970), 479-495. [54] I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Scott, Foresman, and Co., Glennview, IL (1967). [55] S. Smale, Genralized Poincare's conjecture in dimension > 4, Ann. of Math. (2) 74 (1961), 391-466. [56] J. Stallings, The piecewise linear structure of euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481^88. [57] J. Stallings, The embedding of homotopy types into manifolds. Preprint. [58] D. Sullivan, Triangulating homotopy equivalences. Doctoral dissertation, Princeton Univ. (1966). [59] D. Sullivan, On the Hauptvermutung for manifolds. Bull. Amer. Math. Soc. 73 (1967), 598-600. [60] C.T.C. Wall, Homotopy tori and the annulus theorem. Bull. London Math. Soc. 1 (1969), 95-97 [61] C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, London (1970). [62] C. Weber, Velimination des points doubles dans le cas combinatoire. Comment. Math. Helv. 41 (1966-67), 179-182. [63] J.H.C. Whitehead, Simplicial spaces, nuclei, and m-groups, Proc. London Math. Soc. 45 (1939), 243-327. [64] H. Whitney, The self-intersection of a smooth n-manifold in In-space, Ann. of Math. (2) 45 (1944), 220246. [65] B.C. Zeeman, Unknotting spheres, Ann. of Math. (2) 72 (1960), 350-361. [66] B.C. Zeeman, Seminar on Combinatorial Topology (notes), I.H.E.S. (Paris) and Univ. of Warwick (Coventry) (1963-1966).

This Page Intentionally Left Blank

CHAPTER 6

Geometric Group Theory^ J.W. Cannon Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA E-mail: cannon @ math. byu. edu

Contents 0. Introduction 1. Paradigmatic work in geometric group theory 1.1. Dehn's work on the fundamental group of a topological surface 1.2. Gromov's theorem on groups of polynomial growth 1.3. Stallings's theorem on the number of ends of a group 1.4. Mostow's rigidity theorem 1.5. Thurston's geometrization conjecture 2. The foundations of geometric group theory 2.1. Cayley graphs or Dehn Gruppenbilder 3. Examples 3.1. Examples of finitely presented groups 3.2. The standard geometries in dimensions 2 and 3 3.3. The Cayley graphs of the examples and the geometries with which they are associated 4. Abstract classes of groups 4.1. Word hyperbolic or negatively curved groups 4.2. Automatic groups 4.3. CAT(O) groups 5. Other geometries to explore 6. Other geometric invariants References

*This research was supported in part by an NSF research grant. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

261

263 264 264 265 266 267 267 268 273 283 283 286 289 297 298 300 302 303 303 304

Geometric group theory

263

0. Introduction Almost twenty years ago, W.R Thurston established a firm link between classical geometry and low-dimensional topology in dimension 3. The basic reference is [47]. These notes are currently available from the Geometry Center at the University of Minnesota in Minneapolis. The first four chapters of the notes have been carefully reworked and expanded and are available in book form [48]. Thurston's work made it very clear that the dominant invariant in the study of 3-manifolds is the fundamental group and that the nature of the fundamental group is very much a function of the geometry associated with both the group and the manifold through his work. His results suggested that geometry might serve as a guiding principle in the study of infinite group theory. Such thoughts were, of course, not entirely new, for combinatorial group theory had largely begun because of topological and geometric work of Max Dehn early in the century. The fundamental reference is [20]. However, combinatorial group theory had largely abandoned, or at least somewhat obscured, its topological and geometric roots in an attempt to distill the combinatorial essence of the arguments. Thurston's work brought renewed emphasis to those roots. The hopes raised by Thurston's theory, that concentration on geometry might enrich infinite group theory once more, have been amply borne out during the two decades intervening now between Thurston's original work and the present. There has been a veritable explosion in the application of geometric methods to infinite group theory. This chapter is intended as an entry point into the resulting theory. Section 1 is motivational and outlines some of the paradigmatic work in geometric group theory. Section 2 explains, or gives specific reference to, the foundational concepts connecting infinite group theory and geometry. In order to feel at home in geometric group theory, one must come to understand at least the following concepts: • What is a group presentation? What are generators and relators? • How does a presentation define a group? • How does one obtain a presentation for the fundamental group of a space? • What is a geometry? • What is a geometric action of a group on a geometry? • What is the Cayley graph of a group? • How does one construct a Cayley graph by Todd-Coxeter coset enumeration? • In particular, what is a free product with amalgamation, what is an HNN extension, and how does one construct their Cayley graphs? • If a group acts geometrically on a geometry, how uniquely is the geometry determined? (Answer: the geometry is unique up to quasi-isometry.) Section 3 is dedicated to the proposition propounded by Marshall Hall: "Theorems change, but examples remain." In Section 3.1 we give a family of examples of finitely presented groups naturally associated with geometries. In Section 3.2 we then describe in very abbreviated fashion the prototypical, standard geometries in dimensions ^ 3 with which the sample groups are associated. Then finally in Section 3.3 we explain how to construct the Cayley graphs of most of the groups in Section 3.1. We would hope that the

264

J.W. Cannon

reader would at first ignore Section 3.3 and attempt to construct the Cay ley graphs with the help only of the material in Section 2. To do so would be a difficult, lengthy, perhaps impossible exercise; but, if carried out with even partial success, it would firmly place the reader in the subject. After one has the appropriate Cayley graphs, one can use them as discrete models for the geometries which they approximate. Much can be learned about the geometries by thinking about the associated Cayley graphs. In fact, the author's own entry into the field came from his own self-imposed task of constructing "all" of the Cayley graphs of the groups described in the wonderful book [18]. Though we did not succeed in constructing all of these Cayley graphs, we did observe fascinating things about groups which have served as the basis for much of our work. Section 4 outlines three of the most important abstract classes of infinite groups studied by means of geometric group theory, the word-hyperbolic or negatively curved groups, the automatic groups, and the CAT(O) groups. Section 5 enumerates some of the geometries one might study as motivation for further work in geometric group theory. And finally. Section 6 directs the reader toward the existing work on the numerous topological and geometric invariants that have application to the study of infinite groups.

1. Paradigmatic work in geometric group theory Every scientific subject is driven by its best and most suggestive results. In this section we describe five of the many beautiful results of geometric group theory which have been extremely influential in setting the direction of this subject.

1.1. Dehn 's work on the fundamental group of a topological surface Max Dehn [20] introduced the three fundamental problems of combinatorial (and geometric) group theory. They are of course, the word problem, which Dehn called the identity problem, the conjugacy problem, which Dehn called the transformation problem, and the isomorphism problem. Here are Dehn's definitions. Suppose an element of the group is given as a product of generators of the group. One should give a method in order to decide in a finite number of steps whether this element represents the identity element of the group or not. T H E WORD PROBLEM.

T H E CONJUGACY PROBLEM. Suppose two elements S and T of the group are given. A method is sought to decide the question whether S and T are conjugate in the group; that is, does there exist an element U in the group such that S = UTU~ ^ ? T H E ISOMORPHISM PROBLEM. TWO

are isomorphic or not.

groups are given. One should decide whether they

Geometric group theory

265

Dehn gave beautiful solutions to all three problems when the groups involved are the fundamental groups of compact 2-dimensional surfaces. We shall only describe Dehn's solution of the word problem, and only that in the case where the surface is closed, orientable, of genus 2. The fundamental group has the following presentation. G = {xuyuX2,y2

\xiyix^^y^^X2y2X2^y2^

= l).

We call the long relator, which is the product Ui, >'i][jC2, }^2] of two commutators, R. We consider all cyclic permutations of R and its inverse as well as all trivial relators of the form z ' z~^, where z can be any of the generators or their inverses. We call the resulting finite collection of words the set S of fundamental relators for G. Dehn's algorithm for solving the word problem is the following. Let w be any word in the generators of G or their inverses. Look in w for an occurrence of a word M such that one of the elements of S has the form Mm, where M is a word that is longer than the (possibly empty) word m. Replace the occurrence of M by the word m~^. Continue until no more such shortening replacements can be performed. Then w represents the identity element of G if and only if the resulting maximally shortened word is the empty word. Dehn's solution to the conjugacy problem used his solution to the word problem and had a similar flavor. Dehn's solution to both problems used his familiarity with non-Euclidean geometry and the fact that the universal cover of his surface was in fact the non-Euclidean plane. Dehn's solutions were generalized in two directions. First of all, the combinatorial underpinning of the argument was distilled; the consequence was small cancellation theory. A fundamental reference is [32, Chapter V]. In the late 1970's we discovered that Dehn's algorithms could be generalized almost verbatim to all groups acting nicely on non-Euclidean geometry, and in fact to any groups that acted sufficiently like non-Euclidean hyperbolic geometry. The reference is [11]. Gromov gave an appropriate definition of the class of groups acting sufficiently like non-Euclidean geometry in the fundamental paper [29]. Gromov showed that small cancellation theory could be made very geometric and returned the subject to the geometric grounds where it began.

1.2. Gromov's theorem on groups ofpolynomial growth Consider the free Abelian group Z 0 Z on two generators and the free non-Abelian group Z * Z on two generators. If one expresses each element as a word of minimal possible length in terms of the standard generators and their inverses, then the free Abelian group has exactly An elements of minimal length n for n^ I, while the free non-Abelian group has exactly 4 • 3"~^ elements of minimal length nforn^l. We say that the free Abelian group has polynomial growth since there is a polynomial in n which bounds the number of elements of length n and that the free non-Abelian group has exponential growth because there is no such polynomial and in fact the number of elements is bounded below by an exponential function of n. Gromov has proved the following theorem [28]:

266

J.W. Cannon

THEOREM. A finitely presented group G has polynomial growth if and only if G has a nilpotent subgroup of finite index.

In addition to its intrinsic beauty, the result and its proof are of interest to topologists and geometers for at least three reasons. (1) The result shows that the simplest of geometric properties, namely the geometric growth rate of a group, can have profound influence on the algebraic structure of a group. (2) Gromov describes how a sequence of metric spaces can converge to another metric space. (3) Gromov uses, at a critical juncture, the solution to Hilbert's fifth problem about the topological structure of Lie groups.

1.3. Stallings's theorem on the number of ends of a group Let X denote a locally compact, connected metric space. If A' is a compact subset of X, then we say that a component C of X \ A' is unbounded if the closure of C in X is noncompact. We define e(X, K) to be the number of unbounded components. If K' is a compact set which is larger than K, then each unbounded component of X \ ^ will contain at least one unbounded component oiX\K' so that e{X, K') ^ e{X, K). We define the number of ends of X to be the supremum over all K of the numbers e{X,K). The reader will easily verify that the real line has 2 ends, the plane and all Euclidean spaces of higher dimension have only one end. One can assign a number of ends to a finitely generated group in two steps. First one assigns to the group and any of its finite generating sets a locally compact, connected metric space, namely its Cayley graph (see Section 2.3). The Cayley graph has a number of ends by the previous paragraph, and an easy argument shows that the number of ends does not depend on the generating set. We call this the number of ends of the group. For the free-Abelian group of rank 2 with its usual two-element generating set, the Cayley graph is the square unit lattice in the Euclidean plane; it clearly has 1 end. For the free non-Abelian group of rank 2 with its usual two-element generating set, the Cayley graph is an infinite tree with four edges meeting at every vertex; it clearly has infinitely many ends. It has long been known that a group has either 0,1,2, or infinitely many ends. The groups with 0 ends are finite. The generic case seems to be the groups with 1 end. Stallings [45] characterizes groups with 2 or infinitely many ends as follows. A finitely generated group G has exactly two ends if and only if G has an infinite cyclic subgroup of finite index. A finitely generated group G has infinitely many ends if and only ifG can be factored in one of the two following ways: (1) G is a free product with finite amalgamating subgroup where this amalgamating group is properly contained in both factors and of index > 2 in at least one factor, (2) G is an HNN extension amalgamated over a finite subgroup which is properly embedded in the base group.

THEOREM.

Geometric group theory

267

(Free products with amalgamation and HNN extensions, and their Cayley graphs, are described at the end of Section 2.) Stallings remarks that he discovered the proof of this theorem by contemplating Papakyriakopoulos's proof of the sphere theorem.

1.4. Mostow's rigidity theorem It is sometimes possible to endow a topological manifold with a Riemannian metric of constant curvature. For example, a torus of any dimension can be endowed with a flat Euclidean metric. A 2-dimensional surface of genus g at least two can be endowed with a Riemannian metric of constant curvature — 1; as a consequence, the universal cover of such a surface is naturally isometric with non-Euclidean hyperbolic geometry and the covering translations within this geometry are hyperbolic isometrics. In dimension 2, the dimension we have just been discussing, there is a large space of such metric structures. This space is called Teichmiiller space. It is homeomorphic with Euclidean space of dimension 6g — 6. This space has been studied extensively in classical complex variables and conformal mapping. In higher dimensions however Mostow noted that such metrics of constant negative sectional curvature occur much less frequently and then are much more rigidly determined. Here is a weak version of Mostow's rigidity theorem [38]. MoSTOW RIGIDITY THEOREM. Suppose that M\ and M2 are closed manifolds of dimension n ^ 3 endowed with Riemannian metrics of constant sectional curvature equal to —1. If Ml and M2 have isomorphic fundamental groups, then Mi and M2 are isometric. Thus all of the differential geometric invariants of Mi {volume, spectrum of geodesic lengths, conformal structure at infinity, etc.) are in fact topological invariants of Mi. The proof makes strong use of the space at infinity, quasiconformality, ergodicity, weak metric equivalences called quasi-isometry, all of these notions being strongly geometric. The theorem suggests that in dimensions at least 3 algebra may in appropriate settings determine topology, and even Riemannian geometry, completely.

1.5. Thurston's geometrization conjecture The geometric flexibility of topological surfaces in dimension 2 suggests that geometry will not play a major role in the study of manifolds. Mostow's rigidity theorem is a first corrective in showing that, in dimensions higher than 2, algebra, topology, and geometry can be very tightly associated. Thus Thurston could say approximately the following in his address to the International Congress of Mathematicians in Helsinki in 1978: "A compact manifold is nearly determined by its fundamental group. We can almost identify the geometric study of manifolds with the algebraic study of finitely presented groups." Thurston analyzed the geometric structure of Haken 3-manifolds (his results have since been generalized to a larger class of 3-manifolds). He identified all of the geometries

268

J.W. Cannon

(simply connected, complete Riemannian manifolds, homogeneous under isometry) which could serve as universal covering space for a closed 3-manifold with Riemannian metric in such a way that the covering transformations were by isometry. There were exactly eight. (We shall describe them in a later section, Section 3.2.) He then showed that every Haken 3-manifold had a canonical decomposition into pieces each of which admitted a complete Riemannian metric of finite volume with universal cover isometric to one of the eight geometries. We summarize this result in the following abbreviated fashion: THURSTON' S GEOMETRIZATION THEOREM.

Every Haken 3-manifold admits an essen-

tially unique geometric structure. One good reference to the Thurston work is [37]. The Thurston work is explicitly about topology, but it, together with Mostow's rigidity theorem, suggests firm connections with group theory and outlines implicitly a path to be taken in understanding the groups which arise in 3-dimensional topology, namely to analyze the individual groups associated with the 8 geometries and how those groups can be assembled to obtain all 3-manifold groups.

2. The foundations of geometric group theory In this section we address the fundamental issues that must be understood if one is to be comfortable with geometric group theory. These issues were outlined point by point in the introduction. • What is a group presentation? What are generators and relators? A group presentation has the form G = {C \ R).ln this presentation, C denotes a set of symbols or letters called the generating set for the presentation. Each letter c G C is paired with another letter c~^ called its inverse letter. Sometimes we assume for convenience that C = C~^; at other times we simply describe C by giving only one element of each element-inverse pair. In either case, a word in the alphabet C is a finite sequence of elements of C U C~^ The set R is a. collection of words in the alphabet C and is called the defining relator set for the presentation. The symbol G refers to the group defined by the presentation as described in the next section. Sometimes it is convenient to define a relator set by equations of the form v = w, where v and w are words; this equation, which is called a relation rather than a relator, is to be associated with the relator v~^ -w. That is, relations are associated in a natural way with relators. • How does a presentation define a group? If / : C U C~^ ^- / / is a function into a group H such that, for each c G C U C~^ we have f{c~^) = / ( c ) ~ ^ then there is a natural extension of / which takes the collection of all words in the alphabet / into the group H\ one simply takes a word c\ • C2 • • • c^ to the element f{c\) • /(c2) • • • /(c„). We also denote the extended map by / . We consider the category whose objects are functions / : C U C~^ -^ if, as in the previous paragraph, but with the additional property that the extended function / sends each word in the defining relator set R to the identity element of H. We define a morphism in

Geometric group theory

269

this category from object f\:C U C~^ ^^ H\ to object / 2 : C U C~^ ^- //2 to be a group homomorphism h.Hx^- H2 such that, for each c G C U C~^ h{f\ (c)) = /2(c). The group defined by the presentation G = (C | /?> is any initial object in this category; we recall that an initial object of a category is an object which admits a unique morphism into any other object of the category. Initial objects are clearly unique up to isomorphism. Here is a precise construction of the group defined by the presentation G = {C \R). Let W denote the set of all words in the generating set C. Let T denote the set of trivial words, that is words of the form c - c~^, where c G C U C~^ Declare two words w and w' to be equivalent if w^ can be obtained from w by either inserting or deleting at some point a copy of a word from RUT. Extend this notion of equivalence to an equivalence relation ~ . Let G = W/^ denote the set of equivalence classes. Define the product of two equivalence classes [w], represented by the word w eW, and [x], represented by the word x eW,to be the class [wx] represented by the concatenation wx of w and x. It is easy to check that multiplication is well-defined, defines a group structure on G, and that the function / : C U C~^ -^ G which takes c to [c] is an initial object in the category defined in the preceding paragraphs. A word equivalent to the empty word is called a consequence of R. • How does one obtain a presentation for the fundamental group of a space? We assume that the reader is quite familiar with the fundamental group. A good basic reference is [35]. It is easy to obtain a presentation for the fundamental group of a connected space if it happens to be the underlying space of a simplicial complex. Indeed, assume that X is such a space, triangulated, with base point XQ a vertex of the triangulation. Collapse a maximal subtree of the 1-skeleton to the base point XQ. Then the image of the 1-skeleton of Z is a bouquet of loops, each loop supplying a generator for the fundamental group. In the process, each 2-cell is attached along its boundary to the bouquet by an attaching map which can be realized as a word in the generators and their inverses. These words supply the defining relators of a presentation. If the complex is finite, then the presentation is finite. If a space has reasonably simple local structure, then it is likewise easy to obtain a presentation for the fundamental group. We start with a space X that is connected and locally path connected. We say that an open cover is 2-set simple if each element is connected and each loop which lies in the union of any 2 sets of the cover is contractible in the space. We say that the space X is 2-set simple if it has an open cover which is 2-set simple. One could also define the notion of l-set simple space; such a space is called semi-locally simply connected. Semi-local simple connectivity is important as the condition necessary and sufficient for the existence of a universal covering space. We shall see in the next theorem that it is easy to find a presentation for the fundamental group of a space that is 2-set simple. The result will make use of the notion of nerve of a covering. If L^ is a covering, then the nerve N{U) has one vertex v{u) for each element u eU. A set {v(uo), v(u\),..., v(uk)} of /: + 1 vertices spans a ^-simplex of the nerve if and only if the corresponding elements uo, ...,Uk have nonempty intersection. THEOREM. Suppose X is a space that is connected and locally path connected, with base point xo, suppose X is 2-set simple, and suppose U is an open cover of X which is 2-set

270

J.W. Cannon

simple. Then the fundamental group of the nerve N(U) of the open covering U is isomorphic with the fundamental group ofX. COROLLARY. If M is a compact metric topological manifold with boundary, then the fundamental group is finitely presented.

It is clear that M has a finite open covering which is 2-set simple. The nerve is a finite simplicial complex, therefore has finitely presented fundamental group. D PROOF.

Pick for N(U) a base vertex v(uo) where XQ e WQ ^ U. We establish the following facts in turn. (1) Loops in the nerve N(U) can be reasonably easily mapped to loops in X. (2) The image of a loop is well-defined up to homotopy and so has well-defined image in the fundamental group of X. (3) A nullhomotopic loop maps to a nullhomotopic loop; consequently the map on loops is well-defined on homotopy classes of loops in N(U) and defines a homomorphism from fundamental group of N(U) to the fundamental group of X. (4) The homomorphism is surjective and has trivial kernel, hence is an isomorphism. (1) Let P denote the set of closed edge paths in N(U) based at I;(MO)- For each loop p e P we construct a loop ^ in X based at xo as follows. Represent p by a. sequence (v(uo), v(ui),..., v(un), v(uo)) of vcrticcs defining the endpoints of successive edges of p. For each /, pick yi e ut, with yo = ^o- Since v(ui) and fCwz+i) span an edge of N(U), the sets w/ and M/+I intersect, so we may take a path qi in their union which joins yt to y/+i. We then define q to be the path which is the concatenation qoqi • • • ^„ of the paths qt. (2) We claim that the homotopy class of the path q is well-defined in the sense that it does not depend on the choices of points yi e ui and paths qt. Suppose that y- and q- are other admissible choices. Choose paths n in M/ joining yi to y-. Then the paths q[r^j^^q^ n lie in w/ U M/+I, hence are nullhomotopic in X since the cover U is 2-set simple. These nullhomotopies, taken together, build a homotopy from qoq\ •••qn to q^q^ -- -q'^, so that the homotopy class of q is well-defined. (3) Suppose that p is nullhomotopic. Then we see that q is also nullhomotopic as follows. Let g: B^ -^ N(U) be a. singular disk bounded by the nullhomotopic edge path p. We may assume that the domain disk B^ is triangulated and that g is simplicial. We may view ^ as a map from S^ = dB^ into X with S^ divided into segments exactly as in the map g and with segments corresponding to edges of p mapping to paths qi. Our task is to extend this map to all of B^ as a map h: B^ -> X. We extend the map to the entire 1-skeleton of B^ in the same way that we defined ^ in (1). It remains therefore only to define the map h on the 2-simplexes of B^. We therefore consider a 2-simplex i;i 1)2^3 of B^. Since g is simplicial, if w/, with / = 1,2, 3, satisfies g(vi) = v(ui), then the three sets w/ intersect. We map the barycenter b of the 2-simplex to a point in the intersection. We then map the segment bvi to a path in ui which joins h(b) and h{vi). The three segments bvt divide the 2-simplex into three 2-simplexes, with the boundary of each mapped into the union of two elements of U. Since the cover U is 2-set simple, each of these three boundary maps is contractible. Hence we may extend h to each of the three 2-simplexes. This completes the definition of h. PROOF OF THE THEOREM.

Geometric group theory

211

We conclude that the homotopy class of q depends only on the homotopy class of p so that we have an induced mapping / : 7 T \ ( N ( U ) , UQ) -^ 7t\(X, XQ). We also conclude that this mapping is a homomorphism. (4) Suppose that h:S^ ^^ X is SL loop based at the base point XQ. Partition the loop h into subpaths qo,qi, ...,qn, each qt having image in a single element w/ of U, with MQ = ^n being the base element of U. Then p = (v(uo), v(ui),. ..,v(un))isa. loop whose image is homotopic to h. Therefore, the homomorphism / is onto. Suppose now that a loop p is given, with image loop q. Suppose that the path q is nullhomotopic as demonstrated by a map h:B^ -^ X with boundary q. We wish to show that p is also nullhomotopic. It will follow that the homomorphism / is 1 to 1. It is, of course, permissible to change /? by a homotopy and thereby to permit a slight simplification of q. Our aim is a normal form for p such that each edge is either degenerate in N(U) with (possibly nondegenerate) image qi in X which maps into a single element of U or is nondegenerate in N(U) with image qt that is degenerate (constant) in X. We can attain that goal by the following simple stratagem. Replace the edge v(ui)v(ui^{) in p by three edges, v(ui)v(ui), v(ui)v{ui^i), and I;(M/+I)I;(M/+I), the first and the last of these three being degenerate edges. However, in defining q, map the first degenerate edge to a path in ui from yt to a point Zi in the intersection of ui and w/+i, map the nondegenerate edge to the constant path at Zi, and map the final degenerate edge to a path in M/+I from Zi to y/+i. We may therefore, in particular, assume that each edge of q is contained in a single element of U. Since each edge of q is contained in a single element of f/, we may triangulate B^ without changing the boundary triangulation in such a way that each simplex maps into a single element of U. We now define the nullhomotopy g:B^^^ N(U) of p simplex by simplex. An interior vertex z of B^ has image h{z) G X which lies in some element u of U. Define h(z) = v(u), where v(u) is the vertex of N{U) corresponding to w. Let e = v\V2 denote an interior edge of B^ and let b denote its barycenter. The image h{e) lies in a single element u of U. Define g(b) = v(u). Since u contains both h{v\) and h(v2), u intersects the elements of U corresponding to both g{v\) and g{v2)- Hence both g{b)g{v\) and g(b)g(v2) are edges of N(U) so that we may extend g to take the half-edges bv\ and bv2 onto these edges of NiU). Let t = v\V2V3 denote one of the triangles of B^ and let b denote its bary center. The image h(t) lies in a single element u of U, Define g{b) = v(u). Since u contains all three ofh(vi), h(v2), and h{vi,), it follows that u intersects the elements of U previously chosen with respect to all the faces of t. Consequently, the entire boundary of t is already mapped by g into the star of g(b) in N(U). We may thus extend g to all of r by a cone map. This completes the construction of g:B^ -^ N(U) which shrinks the closed path p and completes the proof of the theorem. D • What is a geometry? For our purposes, a geometry is a topological space endowed with SL proper path metric; a path metric is a metric such that the distance between each pair of points is realized as

272

J.W. Cannon

the length of some path in the space joining those points; the metric is proper if closed metric balls of finite radius are compact. What are the prototypical geometries? There is one standard geometry in dimension 1, namely the real line R with its Euclidean metric d{x, y) = \x — y\. There are three standard geometries in dimension 2, and eight in dimension 3; we shall enumerate them in Section 3.2. We consider them prototypical because (1) They are simply connected, connected, metrically homogeneous Riemannian manifolds. (2) They admit a geometric action (defined immediately hereafter) by some group. • What is a geometric action of a group on a geometry? An action G x 5 —> 5 of a group G on a set 5 is a function {g,s) \-^ gs such that, for each g\,g2 e G and for each s e S, g\{gis) = (^1^2)'^ and id • 5 = ^. An action is isometric if 5 is a metric space and, for each g e G and for each x,y e S, d(gx, gy) = d(x,y). An action is cocompact if the orbit space S/G is compact. An action is properly discontinuous if, for each compact set K C S, the set {g G G | ^ fl gK 7^ 0} is finite. An action is geometric if 5 is a geometry and the action is isometric, cocompact, and properly discontinuous. One should view the defining properties of a geometric action in the following way. The isometry condition connects the group action with the metric. The compactness condition ensures that the group is relatively large with respect to the size of the geometry. The proper discontinuity property ensures that the group is not too large with respect to the size of the geometry. That is, the group is compatible with the geometry and is neither too small nor too large, but rather just right. We shall see in Section 2.4 that all geometries on which a group can act geometrically are equivalent under the equivalence relation of quasi-isometry. Suppose now that X is a compact geometry, not simply connected, but semi-locally simply connected (1-set simple in our previous terminology, consequently also 2-set simple since the space is metric, with finitely presented fundamental group). Then we may lift the local metric on X to a path metric on the universal covering space XofX. Then X becomes a geometry. The fundamental group acts on this geometry. The action is isometric precisely because the metric on X is lifted from a metric on X. The action is cocompact because the compact space X is the quotient of X under the action. The action is properly discontinuous by a fundamental property of covering transformations. That is, the finitely presented group TTi (X) (base point suppressed) acts geometrically on Z. ^ Properties of the group n\{X) are reflected in properties of the geometry X. Here is a sample well-known theorem. A group G acts geometrically on an n-connected geometry, n^O, if and only if there is an Eilenberg-MacLane space K{G,l) with finite (n + I)-skeleton.

THEOREM.

The case n = 0 implies that a group must be finitely generated in order to act geometrically on a geometry. The case n = I implies that a group must be finitely presented in order to act geometrically on a simply connected geometry. And so forth. One proof of the theorem appears in the appendix of the paper [13]. • What is the Cayley graph of a group?

Geometric group theory

273

2.1. Cay ley graphs or Dehn Gruppenbilder Let G denote a group and C = C~^ a generating set for G. Then the Cayley graph or Dehn Gruppenbild F = r(G, C) for G with respect to the generating set C is a 1-dimensional CW complex defined as follows. The vertices of F are the elements of G. If i; is a vertex and if c G C, then we associate with v and c a labelled and directed edge denoted (v,c,V'C) which has v as its initial endpoint, i; • c as its terminal endpoint, and c as its color or label. The directed edge (v,c,v • c) is identified topologically with a second edge (i; • c, c~^ v) in an order reversing fashion. The second edge is considered the inverse or reverse of the first. The two directed edges define the same single undirected topological edge joining v and V ' c. The resulting graph can be turned into a path metric space by assigning a length of 1 to each edge. There is a natural action of G on its Cayley graph F. Here is the action on vertices v and edges (v,c,v - c), respectively:

{g, (v, c, V 'c))\-^

(gv, c, gv • c).

By assigning a linear structure to one edge in each orbit of edges and transferring that structure to all edges in the orbit by the action, we obtain a path metric that is invariant under the action. THEOREM. The Cayley graph F is a geometry if and only if the generating set C is finite. If the Cayley graph F is a geometry, then the natural action G x F ^^ F is a geometric action.

Note that this result is compatible with the result of the preceding subsection where it was noted that a necessary and sufficient condition that G act geometrically on some (connected) geometry is that the group G be finitely generated. Topologists may find the following alternative description of F enlightening. Pick a presentation fov G: G = {C \ R) where C = C~^ is the given generating set for G and the set R is a. defining set of relators for G. Choose some point b as base vertex for a CW complex. Attach to Z? a bouquet B of loops, one for each element-inverse pair c, c~^ from C. For each relator r e R (r being a finite word in the generators C) attach a disk D(r) to B so that the boundary circle of D(r) has image in B which traces out the word r in B. Let K denote the resulting 2-dimensional CW complex. Then the fundamental group of ^ is G. Let K denote the universal cover of K. Then K is called the Cayley complex of the presentation, and its 1-skeleton is the Cayley graph F = F(G, C). The alert reader will note that there is a slight incompatibility in our two descriptions of the Cayley graph when one of the generators c G C is its own inverse. In the graph description one obtains edges of the form (v,c,v - c) with inverse edge (i; • c, c, v), the pair forming a single undirected edge. In the topological universal-cover description one obtains two edges spanned by a disk which gives the relator c^ = id. Collapsing such disks to an arc recovers the graph description from the topological description.

274

J.W. Cannon

Fig. 1. Soccerballohedron, the Cayley graph of the alternating group on five letters.

As an illustration without supporting proofs we take the group defined by the presentation

Here is the Cayley complex with disks collapsed corresponding to the relator jc^ = 1; the topological space underlying the Cayley complex is the 2-sphere S^ which we view as the 1-point compactification of the Euclidean plane, with the compactifying point at infinity. The 1-skeleton naturally subdivides the 2-sphere into twelve pentagons (shaded, the outermost one containing the point at infinity) swimming in a sea of twenty hexagons. The pentagons correspond to the relator y^ = 1, the hexagons represent the relator (xy)^ = 1, and the edges connecting different pentagons represent the edges to which the relators labelled x^ = 1 have been collapsed. The configuration pictured above has been enjoyed for millenia. The sports enthusiast will immediately recognize the pattern as being that of the soccer football, or, as it is affectionately known, the soccerballohedron. The most striking conclusion to be drawn from the figure is that the Cayley graph of a group, realized geometrically, has a natural (in this case, spherical) shape. What information can be read easily from this geometric representation? Order: there are 60 vertices so that the group has order 60. Commutativity: start at a vertex and follow paths labelled xy, then yx; the two paths end at different vertices; hence the group is non-Abelian. Cosets: the twelve pentagons represent the twelve left cosets in G of the subgroup {y). Multiplication table: a standard multiplication table would require 60^ = 3600 entries; all of that information is encoded in the graph and its directed adjacencies. Symmetry: every vertex sits in the graph exactly in the same manner of very other vertex; it is therefore irrelevant which vertex is considered the identity vertex.

Geometric group theory

275

Recognizing the Cayley graph. In giving the sample Cayley graph above, we have obviously ignored a number of critical issues. How does one construct the Cayley graph of a group? When one encounters a graph, how does one recognize that it is in fact a Cayley graph of some group? that it is the Cayley graph of the group in which we are interested? We explain these issues in this subsection. The most daunting result states that it is, in general, impossible to construct a prescribed finite chunk of an infinite Cayley graph algorithmically; this impossibility result is known as the unsolvability of the word problem for finitely presented groups. Two enlightening references concerning this result are [43, Chapter 12, pp. 277-321], [32, Chapter IV, Section 7]. Nevertheless, many Cayley graphs can be constructed algorithmically, and there is a natural algorithm called Todd-Coxeter coset enumeration which works for all groups in the sense that it will construct each finite chunk of the Cayley graph correctly in a finite amount of time. The difficulty is that, in general, one does not know when to stop and say that the algorithm has succeeded; the algorithm grinds away and constructs larger and larger chunks of the graph correctly, but it takes so much time that no computable sequence of stopping times grows large enough rapidly enough to serve as a predictor which says when enough is enough. We outline here the theoretical underpinnings which ensure the correctness of the ToddCoxeter algorithm and which allow one to construct and recognize the Cayley graph of many finitely presented groups. In fact, as we shall see in a later section, the Todd-Coxeter algorithm succeeds nicely for the generic finitely presented group, namely for all negatively curved (that is, word-hyperbolic) groups (Section 4.1). All of the results which we cite in this section are elementary. Readers who will develop proofs for each of them will soon find themselves completely comfortable with Cayley graphs. The proofs appear in an unpublished expository paper of the author [14] which he will be happy to supply to interested parties. We give a few of the proofs here as hints to the reader who is attempting to work out the proofs alone. A colored graph F with color set C = C~^ consists of a set V of vertices and a set EcVxCxVof directed and labelled edges (v,c,w) such that, if (v,c,w) is an edge, then its inverse or reverse (w,c~^ ,v) is also an edge. This inverse pair of edges is said to define the same topological undirected edge. The standard example of a colored graph is the coset graph (Schreier or Todd-Coxeter diagram): let G denote a group, C = C~^ a generating set, and H a subgroup of G; set r = (V, C, E) where V = {Hg\ g e G} md E = {(Hg, c,Hg - c) \ g e G, c e C). If H is the trivial subgroup of G, then F is the Cayley graph. What properties of colored graphs single out the coset graphs and Cayley graphs? The connectedness property. Since C is a generating set for G, coset and Cayley graphs are clearly connected: there is a path of edges labelled by appropriate generators which take any element of G (or coset of H)io any other element of G (or coset of / / ) . We call a graph which satisfies the connectedness property a connected graph. The permutation property. Since multiplication (Hg, c) \-^ Hgc by c permutes the cosets of H, there is one and exactly one edge labelled c entering and exiting from each vertex of F. We call a graph which satisfies the permutation property Si permutation graph.

276

J.W. Cannon

Characterization theorem for coset graphs. A colored graph is a coset graph if and only if it is a connected permutation graph. What distinguishes Cayley graphs among coset graphs? Suppose that we have a colored graph r = (V, C, E) that is a connected permutation graph. Given any finite word w in the generating set C = C~' and any vertex f G V, there is a unique edge path in F which begins at v and whose successive edge labels are the letters of w. We say that if is a relator at i; if this unique edge path is a closed path. The homogeneity property. Since G acts isometrically on its Cayley graph F by left multiplication and since there is an element of G which takes any vertex v to any other vertex v^ (namely, left multiplication by v' • f"^), any relator u; at i; is also a relator at v'. We summarize this by saying that F is homogeneous. Characterization theorem for Cayley graphs. A coset graph is a Cayley graph if and only if it is homogeneous. As defining relators one may take any collection of relators of which all other relators are consequences. At this point we refer the reader back to the section where we constructed the group defined by given generators and relators. Suppose that we have a Cayley graph F. For which group is it a Cayley graph? By the characterization theorem for Cayley graphs, the graph F has the same relator set at each of its vertices, namely precisely those words are relators which are consequences of the defining relator set. The Cayley graph F = (y, C, £•) has as its relator set at each vertex precisely the words that are consequences of the relators R. This property of relator sets characterizes the Cayley graph ofG = {C\R). The observations of the previous paragraph supply the key to the construction of the Cayley graph F for G: we must construct a connected permutation graph such that the relators at each vertex are precisely those words that are consequences of the defining relators R of some presentation G = {C \R). The following lemmas indicate how to control the relators of a constructed graph. T H E BASE POINT LEMMA. Let F denote a connected colored graph and let v and w denote vertices of F. Then every relator at v is a consequence of the relators at w. PROOF. Let r be a relator at v. Let /? be a path from u; to u in the (connected) graph F. Thenpr/?"^ is a relator at w. Then r is equivalent to p~^prp~^p via trivial relators, which is equivalent to p~^ p via the relator pr/?"^ at if, which is equivalent to the empty word via trivial relators. D

Let F denote a connected colored graph that is separated by each of its edges. Then all of the relators in F are consequences of the trivial relators T.

T H E TREE LEMMA.

PROOF. Hint: if r is a relator at vertex u, then the path which begins at v and is labelled by the letters of r has vertices which are furthest from i;; take the first edge leading to such a vertex and show that edge is cancelled by the very next edge. D

Let F\ and F2 denote two disjoint connected colored graphs and let v\ and V2 denote vertices of F\ and F2, respectively. Let F denote the graph formed by

T H E W E D G E LEMMA.

Geometric group theory

277

identifying v\ with f 2. Then every relator in F is a consequence of the union of the relators of Fx and F2. Hint: use the base point lemma to see that one needs only to consider the relators at vi and V2. •

PROOF.

Let F denote a connected colored graph and let ~ denote an equivalence relation on the vertices of F. We may then take a quotient F' of the graph F by taking the equivalence classes as vertices and identifying edges that have the same initial vertex, the same terminal vertex, and the same label. THE QUOTIENT LEMMA. Let R denote a set of words. Let F denote a connected colored graph. Let ~ denote a relation (not necessarily an equivalence relation) such that, if vertices v\ and V2 are related, then there is a word r e R such that either v\ - r = V2 or V2 ' r = v\\ that is, there is a path joining v\ and V2 with label r. Extend ~ to an equivalence relation on the vertices of F. Then every relator of the quotient graph F' is a consequence of the union of the set R and the relators of F. PROOF. Let e\"-en denote an edge path in the quotient graph that is a relator. Lift each edge Ci to an edge // in the original graph F. The terminal vertex of ft is joined to the initial vertex of /;+i (or, of /„ and /o) by a path gi which is labelled by a product of elements of RVJ R~^. The word e\"-en is therefore equivalent io e\f\e2f2"-^ufn via elements of R\J R~^, and the latter word is a relator in T. D

A coset graph F = {V, C, E) is the group graph defined by presentation {C \ R) provided (1) Each r e R is a relator at each vertex of F, and (2) Each relator of F is a consequence of R. THE RECOGNITION THEOREM.

PROOF. We indicate the argument. In order to show that F is the desired Cayley graph, we must show that the relators at each vertex are precisely the consequences of R. By (2) there are no superfluous relators at any vertex. Hence we can finish the proof by showing that every consequence w of the defining set R of relators is in fact a relator at every vertex v of F. Fix a vertex v of F and fix a consequence w of the defining relator set R. Then w can be transformed into the empty word by a finite number of operations which either insert or delete an element of RVJT, where T is the set of trivial words. We may assume by induction (which begins with the empty word) that w is only one transformation away from a word which is a relator at v. Let us suppose, for example, that w = w\ -r •W2 where r e RD T and where u;i • w;2 is a relator at v. Since F is a permutation graph (has the permutation property), every trivial relator is a relator at each vertex of F; taking this fact together with property (1) of the hypothesis, we see that r is necessarily a relator at each vertex of F , in particular at the terminal vertex v - wi of the path which begins at i; and is labelled by the word w\. But this means that we can delete from the path labelled w which begins at v the subpath which begins at f • u;i and is labelled r without disrupting the path.

278

J.W. Cannon

We pass thereby to the path which begins at v and is labelled by if i • u;2. By inductive hypothesis, this path is a closed path (a relator). It follows that the original path labelled by w is also closed. The complementary case where w is formed by deleting a relator r is handled similarly. In this case too we conclude that w; is a relator at v. These observations complete the proof of the recognition theorem. D • How does one construct a Cayley graph by Todd-Coxeter coset enumeration? The Todd-Coxeter algorithm. We give only the details of the construction of the Cayley graph by coset enumeration with the subgroup H being the trivial subgroup. We leave the modifications necessary for / / / 1 to the reader. We assume a group presentation (C | R) given. Our task is to construct a connected permutation graph which has as its relators at each vertex precisely the consequences of R. We proceed by induction. The induction begins with a single vertex which we think of as the identity or base vertex UQ of F . The graph F will be a limit of graphs Fn, each with a base vertex denoted VQ. The limit is taken in the following sense: for each integer A: > 0 there will be an integer n{k) after which the neighborhoods or balls in Fn of radius ^ /: no longer change; then F may be taken as the union of these unchanging balls, individually unchanging but growing with k. The graph Fn^\ is created from the graph Fn by a three-stage process. Stage 1, the expansion stage. In order that F be a permutation graph, we must have at least one edge with each label which emanates from each vertex of T. If this property fails at any vertex of Fn, we call that vertex incomplete. We need to complete each vertex of Fn by adding each missing edge. Each individual edge of course is a tree and so has only trivial relators by the tree lemma. Wedging these edges to Fn adds only consequences of /? U r by the wedge lemma. In the process, of course, we create new incomplete vertices, but that problem will be fixed at the next expansion stage. We call the new colored graph

r.(i). Stage 2, the relator stage. We can assume inductively that the words of R are relators at most of the vertices of r „ ( l ) . However, for some of the more recently added vertices this need not be the case. If i; is a vertex of Fn{l) and if there is a labelled path m Fn{l) which begins at v and which has r G /? as its label, then we may take a quotient graph of Fn{\) which identifies v with f • r. By the quotient lemma, every relator of the quotient graph is a consequence of the relators R. Having carried out this process for every vertex of r„(l) we obtain a colored graph r„(2). As this process is repeated, we ensure condition (1) of the recognition theorem at more and more of the vertices of our graph. Stage 3, the permutation stage. The expansion stage, when repeated, ensures that the trivial relators will be relators at every vertex of our graph. This condition is not, however, sufficient to ensure that the result will be a permutation graph because we may find that we have more than one edge with a given label emanating from the same vertex. We need to fold those edges together so that they become one edge. The vertices which we wish to identify thereby are joined by a path which has a trivial relator as its label; therefore taking such a quotient will, by the quotient lemma, introduce no undesired consequences. We therefore identify vertices which are joined by paths labelled by trivial relators. As a consequence of these identifications we make sure that no more than one edge emanates

Geometric group theory

279

from each vertex with any given label. One should note, however, that the creation of such a quotient may introduce new vertices that have more than one edge emanating from a given vertex with a given label. Thus one may obtain a cascade of identifications. It is in these cascades that the uncertainty in the construction of F occurs. The fact that the word problem cannot in general be solved lies precisely in the existence of these unpredictable cascading collapses. At any rate, after the collapsing has stopped, we obtain a colored graph FfiiS) which we also call F„+i. Note that, for any integer k, there are only finitely many colored graphs of radius ^ k which have no more than one edge emanating from any given vertex with a given label. Hence the balls of radius ^ /: in the graphs Fn, which at worst forn^k may become smaller by the taking of quotients but never become bigger, must eventually stabiHze. Thus the limit F exists. This graph is easily seen, by the recognition theorem, to be the Cayley graph of G. Final remarks on the Todd-Coxeter process. The construction of any infinite Cayley graph by the Todd-Coxeter process takes, of course, infinitely many steps. The remarkable fact is that it is often easy to recognize the algorithmic patterns that occur in the process so that one can see the entire infinite result in clear detail just as one can see the entire unit lattice in the Euclidean plane. The pictures one obtains are almost without exception compellingly beautiful. Exercises on the construction of Cayley graphs EXERCISE 1. Construct the Cayley graph of the soccer ball by the Todd-Coxeter algorithm. Be sure to use the appropriate lemmas and the recognition theorem to verify that you have constructed the right graph. EXERCISE

2. Construct the Cayley graphs of the groups given in Section 3.1.

• In particular, what is a free product with amalgamation, what is an HNN extension, and how does one construct their Cayley graphs? The most useful constructions of combinatorial group theory are free products with amalgamation and HNN extensions. It is a fine exercise to use the tools prepared in the previous section to describe the Cayley graphs of these constructs. One can use the recognition theorem to verify that the descriptions are correct. Thereby one can immediately verify the important normal form theorems for elements of these groups (Britten's lemma for the HNN case). Let G\ and G2 denote groups having isomorphic subgroups H\ and H2. Consider presentations DEFINITION.

Gi =

{Ci\Ri)

for these groups having the special property that in the generating set Q for G/ there is a subset D/ which lies in and generates Hi. Assume further that the sets D\ and D2 correspond bijectively according to the given isomorphism between H\ and H2'. d\(a) G D\

280

J.W. Cannon

corresponding to d2{ot) G D2, where the symbol a is simply an index from some suppressed index set A. Then Xh^ free product with amalgamation has presentation Gi HHx=H2) G2 = (Ci U C2 I /?i U R2, di(a) = d2{ot)). Constructing the Cayley graph of the free product with amalgamation. We must construct a connected permutation graph on the colors Ci U C2 which satisfies the two requirements of the recognition theorem for Cayley graphs. That is, we want the defining relators to be relators at each vertex, but we want every relator in the graph to be a consequence of the defining relators. We build the Cayley graph recursively. We start with a copy of the Cayley graph of G1. The graph is connected and has precisely one edge into and out of each vertex labelled with the colors of Ci. Every relator is, of course, a consequence of the relators R\ and every relator in the defining set R\ is satisfied at each vertex. As defects, however, we do not have edges labelled with the colors of C2, nor have we satisfied the relators of R2 nor the new relators of the form d\ (a) = d2(oi). The principle step of our recursive description deals with a single left coset of H\ in G1. We see these cosets by examining the subgraph with edges labelled by elements of Di. This subgraph falls into components corresponding precisely to the left cosets gH^ of Hi inGi. By homogeneity of the graph, each of these cosets looks in the graph precisely like each of the others and each vertex looks like each of the others as well. We take one coset gH^ and one vertex i; of that coset. Corresponding to that coset, we take one copy of the Cayley graph of G2 and one vertex w of that graph. We identify v and w. By the wedge lemma, all relators in the new graph are consequences of the relators in the old graph. We now take the quotient which identifies the coset gHi with the coset of H2 in the graph of G2 which contains the vertex w. All of the identifications made are consequences of the new relators d\(a) = d2(oi). One obtains a new graph which is a quotient of the two original Cayley graphs, the two graphs intersecting at the vertices corresponding to exactly one coset in each. The resulting graph has been improved in the following way. At each element of the two cosets, both generating sets appear, in and out as required by the permutation property, and all of the defining relators are satisfied. The remaining cosets in the two graphs have not been affected and suffer all of the old defects. However, the recursion now becomes clear. For each incomplete coset of G1 repeat the process with a new copy of the graph of G2. As a consequence, all of the cosets of the original copy of the Cayley graph of G\ will be complete. However, each new copy of the graph of G2 will have cosets that are incomplete. Complete each of those with a new copy of the graph of G1. Now there will be incomplete cosets in the new copies of G1. Complete those cosets. And so forth. Take the union of all the finite stages. This graph satisfies all of the properties of the recognition theorem. Notice that the recursion puts the copies of the graphs of G\ and G2 together in the pattern of a tree. Following paths through the tree in the order of the construction gives a natural description of the elements of the free product with amalgamation.

Geometric group theory

281

DEFINITION. The other construction that is basic is called the HNN extension (after Higman, Neumann, and Neumann). It deals with a single group G which has two isomorphic subgroups H\ and H2. We take a presentation

G =

{C\R)

such that C has subsets D\ generating Hi and D2 generating H2, again in such a manner that the sets D\ and D2 correspond under the given isomorphism of H\ with H2, d\(a) corresponding to d2(a). The HNN extension then has one additional generator /, called a stable letter, and a presentation G*(//j^//2) =(<^'H ^ , td2(a)t-^

=di(a)).

Constructing the Cayley graph of an HNN extension. Again one starts with a copy of the Cayley graph of G and considers a left coset of H\ in G. Corresponding to that (incomplete) coset, one takes another copy of the graph of G. In that new copy, one considers a single left coset of the group H2. One connects the old copy with the new copy via a whole family of edges labelled t which join elements of g\H\ with elements of g2 H2 in a manner corresponding to the isomorphism given between H\ and H2. By homogeneity it does not really matter what vertices are considered as the identity vertex; the result will be the same, and all relators will be consequences of the new relators and the old. One repeats this process with each left coset of H\. However, in the case of the HNN extension we do not thereby complete the graph at this first copy of G; for we have edges labelled t exiting from all of these vertices, but we do not have any edges labelled t entering these vertices. One must also complete the left cosets of H2 in the original copy of the graph ofG. Each new copy of the graph of G has incomplete cosets of both H\ and of H2 which must be completed. Again, after infinitely many steps one obtains the Cayley graph of the HNN extension. Again one notes that it has been put together recursively in essentially a treelike fashion. Tracing paths through the defining treelike structure gives Britten's lemma which supplies normal forms for the elements of the group. Uniqueness of normal forms is obtained by using particular coset representatives for the passage from one copy of the graph of G to another. • If a group acts geometrically on a geometry, how uniquely is the geometry determined? (Answer: the geometry is unique up to quasi-isometry.) Uniqueness of the geometry on which a group acts geometrically. We note that the free Abelian group Z 0 Z on two generators acts geometrically on both its own Cayley graph and on the Euclidean plane. It is clear that these two geometries are not homeomorphic. However, if one stands far back, then the Cayley graph, which is the unit lattice in the plane, looks like the plane, just as a screen door looks Hke a door. That is, in the large the two geometries look alike. We can make this mathematical relationship precise by introducing the notion of quasi-isometry. Two spaces are quasi-isometric if they are Lipschitz equivalent in the large; local differences are ignored. Here are the precise definitions which we take from the discussion in the appendix of [13].

282

J.W. Cannon

DEFINITION. A relation (multi-valued function) R.X -^ Y between spaces X and Y is said to be quasi-Lipschitz if (1) R is everywhere defined (i.e., R(x) ^ 0 for each x € X); and (2) there exist positive numbers, K and L, such that for each A 0 such that d(S o R, idx) < M and d(R o S, idy) < M. DEFINITION. A relation R:X ^^ Y is a quasi-Lipschitz equivalence if there is a quasiinverse S:Y ^^ X such that both R and S are quasi-Lipschitz. If two spaces are quasiLipschitz equivalent, then we say that they are quasi-isometric.

If a group G acts geometrically on two geometries X and Y, then X and Y are quasi-isometric.

EQUIVALENCE THEOREM.

PROOF. The proof is essentially an exercise. For complete details we refer the reader to the reference cited above. However, we do define suitable relations R and S. Fix 6: > 0. Let L^o and VQ be bounded, connected open sets in X and F, respectively, such that the G-translates of Uo and VQ cover X and F, respectively. Let U and V denote the e-neighborhoods of L^o and VQ, respectively. Define R.X^^Y and S:Y -^ X by

R(x) = {yeY\3geG,

xegU,

S(y) = {xeX\3geG.

xegU,

yegV} yegV}.

and

Geometric group theory

283

Note that, as relations, S = R~^. We claim that R and S are quasi-Lipschitz quasi-inverses. Distances are approximated by counting the number of translates of L^ or of V that are crossed in going from one point to another. D 3. Examples Marshall Hall has been credited with the statement, "Theorems change, but the examples remain." In Section 3.1 we give finitely presented groups that act geometrically or almost geometrically on standard geometries. The reader can gain considerable insight by attempting to construct the associated Cayley graphs; by the equivalence theorem at the end of Section 2 these graphs are quasi-isometric to their associated standard geometries; therefore these graphs serve as instructive discrete approximations to the geometries in question. We postpone the description of the geometries to Section 3.2. In Section 3.3, we describe some of the Cayley graphs of these groups and show how the properties of the associated geometries manifest themselves in the combinatorics of the groups.

3.1. Examples of finitely presented groups We attempt to describe the groups by a representative variety of means, and we choose the groups from a representative family of geometries. To the unpractised eye, these groups may look much alike. The point of the list is that they are associated naturally with a large variety of geometries and that they can be fruitfully studied by studying the appropriate geometries. The geometries will be described in Section 3.2. 3.1.1. As our first class of examples we take the free Abelian groups on a finite number n of generators. They have a finite presentation of the form {X\, . . . , Xfi I Xi Xj ^ Xj Xi / ,

or, using the commutator notation [x, y] = {xu...,Xn

xyx~^y~^,

I [Xi,Xj] = l}.

As our second class of examples we take the Fibonacci groups originally defined by John Conway. A number of the relevant references occur in Lyndon and Schupp's book on combinatorial group theory [32, p. 97]. The natural presentation from which the groups received their name are these. Fn = {x\,...,Xn

\XiX2=X3,...,Xn-\Xn=X\,

X„Xi =

X2).

All of these groups are interesting, but we are particularly interested in Fe. Finally we add the group with presentation, G = [a,b,t,A,C\ab

= ba, t^ = l, tat = b, tbt = a, A=a^,

C = ab).

284

J.W. Cannon

3.1.2 and 3.1.3. Dodecahedra and pentagons. The Poincare dodecahedral space is a 3-manifold not homeomorphic with the 3-sphere but whose homology is isomorphic with that of the 3-sphere. One can define the space as an identification of the solid dodecahedron. One identifies opposite faces (in six pairs) by projecting a face through from one side of the dodecahedron to the other and then rotating the face just enough (1/10 of a turn clockwise) so that the two faces coincide. The fundamental group of this manifold is, according to Poincare, not trivial. He discovered this example after making the initial false assertion that all homology 3-spheres are topological 3-spheres. He suggested the fundamental group as an invariant that would distinguish this particular 3-manifold from the 3-sphere and asked (the Poincare conjecture) whether all simply connected, closed 3-manifolds are topological spheres. The Seifert-Weber space is a 3-manifold that is formed again by identifying opposite faces of a dodecahedron, this time with 3/10 of a turn. The right-angled-dodecahedron reflection group is defined in terms of a regular dodecahedron in 3-dimensional non-Euclidean hyperbolic geometry. In that geometry, which will be described in Section 3.2, it is possible to find a regular dodecahedron all of whose dihedral angles are right angles. One then takes the group of non-Euclidean isometrics generated by the twelve reflections in the faces of the dodecahedron. There is a lower-dimensional analog of the group just described. Namely, in 2-dimensional non-Euclidean hyperbolic geometry there is a regular pentagon each of whose angles is a right angle. We may then take the group of hyperbolic isometrics generated by the five reflections in the edges of this pentagon. As groups analogous to the ones just described we include the fundamental groups of the closed, orientable surfaces: Gg = {a\,hx,...,ag,bg\[a\,bx\'-'

[a^, bg^ = l).

We use the notation [a, b] for the commutator aba~^b~K 3.1.4. Knot groups. Consider a polygonal knot (knotted circle) K in Euclidean 3space R^. Project the knot orthogonally into some plane P in such a fashion that any singular point of the projection is a double point where exactly two of the straight segments of K have crossing images where we think of one of the two edges as crossing under the other. The image of ^ is a connected finite graph in the plane consisting of nonsingular arcs joining edge crossings. Form the dual graph by taking one vertex in each complementary domain and joining two of these dual vertices by an edge if the two complementary domains share a nonsingular boundary arc, one edge for each such boundary arc. One can realize each dual edge physically as an edge that starts at the dual vertex in one of the two domains, passes "under" the boundary edge to which it is dual and proceeds to the other dual vertex. If one orients the knot K, and hence the knot projection, then it is possible to orient the dual edges of the dual graph in such a way that, with the preferred orientation, the dual edge passes under its boundary edge from left to right. Now label the dual graph as follows.

Geometric group theory

285

Fig. 2. The trefoil knot and its relator diagram.

Fig. 3. The Fig. 8 knot and its relator diagram.

As one proceeds along the projection of K in the direction of its preferred orientation, one passes from one domain-boundary edge to another. Call two of these edges equivalent if the edge of K which connects them passes over its crossing edge. Extend to an equivalence relation on nonsingular boundary edges. Give two dual edges the same label provided that their corresponding boundary edges are equivalent. We call the labelled dual graph the relator graph of the knot projection. (Actually, the same procedure can be followed out if, instead of a knot K, we take a link in R^.) Associated with the relator graph we have

286

J.W. Cannon

Fig. 4.

a group presentation. We take one generator for each label. We take one relator for each word labelling a closed edge path in the relator graph. All of the relators are actually consequences of any set of relators that form a homology basis for the 1-dimensional homology of the relator graph. We give as examples the trefoil knot group and the figure eight knot group. These presentations are presentations for the fundamental group of the complement of the knot. 3.1.5. We recall the soccer-ball group from the previous section: {x,y\x^

= y^ = {xy)^ = \).

We also consider the direct product of this group with the integers Z. 3.1.6. We consider the group defined by the presentation, P = (xi, yi, X2, y2,t\t

= [x\,y\][x2, yi], t central).

3.1.7. We consider the matrix group A^ which consists of 3 by 3 matrices which are 1 on the diagonal, 0 below the diagonal, and integral above the diagonal. 3.1.8. We consider the group S given by the presentation ia,b,t \ab = ba, tat~^ =b, tbt~

=ab).

3.2. The standard geometries in dimensions 2 and 3 A good reference is [44]. As noted before, a geometry is standard if it is a simply connected, complete Riemannian manifold on which its own isometry group acts transitively on which some group

Geometric group theory

287

can act geometrically and without fixed points. There is one standard geometry in dimension 1, namely the real line R. There are three standard geometries in dimension 2, namely the Euclidean plane, the 2-dimensional Euclidean sphere 5"^, and the non-Euclidean hyperbolic plane H^. Each of these geometries has an analogue in all dimensions greater than 2. Thurston proved that there are exactly eight standard geometries in dimension 3. We shall enumerate the eight 3-dimensional geometries in this section. For some of these geometries we will give a generic description that applies in dimensions greater than 3. 3.2.1. Euclidean «-space R^ has as its points the n-tuples x = (xu ..., Xn) of real numbers and as its Riemannian metric d^^ = djc^ H h dx^. 3.2.2. The Euclidean «-sphere S^ has as its points the vectors x = (jci,..., x„, jc^+i) from Euclidean (n + 1)-space which have Euclidean length 1 and as its Riemannian metric the restriction of the Euclidean metric on /?""^^ to the tangent space of S^. 3.2.3. Non-Euclidean hyperbolic n-space is, of all the standard geometries, by far the most important. Non-Euclidean hyperbolic n-space has a number of standard Riemannian models. A reasonably good short introduction to the properties of this space appears in the paper [16]. Other good references are [5,31,41]. We give here just one of the five standard models for hyperbolic n-space, namely Poincare's upper half space model. As underlying space we take H={(xu...,Xn)eR''\xn>0}. As Riemannian metric we take ds^ =

{dx^-\--"-hdx^)/xl

3.2.4 and 3.2.5. Euclidean geometry is, of course, a product geometry. There are two other standard product geometries in dimension 3. The first is the product S^ x R of the 2-sphere with the line. The second is the product H^ x R of the hyperbolic 2-dimensional plane with the line. In each case, one takes as the new Riemannian metric ds^ = ds^ + ds2, where d^^^ is the standard metric on the first factor, d^l the standard metric on the second factor. 3.2.6. There are three twisted geometries in dimension 3. The first of these is SL2, or the universal cover of the unit tangent bundle to the hyperbolic plane. If we think of the hyperbolic plane in the upper half plane model, then its group of orientation preserving isometrics is the group of Hnear fractional transformations with real coefficients: az + b Z\-^

T'

cz^d

288

J.W. Cannon

with a,b,c, and d real, and with ad —be—I. That is, this isometry group is PSL2(/?). This group of hyperboHc isometrics induces homcomorphisms of the unit tangent bundle. One can extend this group of homcomorphisms by allowing, in addition, rotation of all fibers through some fixed real angle. These rotations commute with the other homcomorphisms we have described and commute as well with one another. That is, we obtain a group of homcomorphisms which has these fiber rotations in the center. After choice of base point, all of these homcomorphisms lift naturally to the universal cover SL2. While in the base space, one cannot distinguish fiber rotations which differ by multiples of In, these homcomorphisms are naturally distinguished in the universal cover since the universal cover unwraps all of the circle fibers into lines. If one chooses the standard Euclidean Riemannian metric at the base point of SL2, then the group of homcomorphisms we have described can be used to pull this metric back to all other points of SL2. We obtain thereby a left invariant metric on SL2. This SL2 is not isometric with H^ x R but it is quasi-isometric with H^ x R. 3.7. The second of our twisted geometries is Nil. As a space it coincides with the space of 3 by 3 real matrices which are 0 below the diagonal, 1 on the diagonal, and arbitrary entries a, b, and c above the diagonal. This space is naturally homeomorphic with 3dimensional Euclidean space with vectors (a,b,c). Matrix multiplication defines a group structure on this space. (a, b, c) • (a, p,y) = (a-\-a,b-\-ay

-\- P,c + y).

If one chooses the standard Euclidean Riemannian metric at the vector (0,0,0), then the group structure may be used to pull the metric back to a left invariant metric at each point. The result is this: ds^ = djc^ + dy^ + (dz - X dyf at the point

(x,y,z)-

3.8. The third and final twisted geometry is Sol. As a space it again coincides with Euclidean 3-space, the set of ordered triples (JC , j , z) with the product topology. A group structure is defined on Sol by the formula (a,b,c)-

{a,P,y) = (a+Qf • exp(-c), Z7 + )S • exp(c),c + y ) .

If one chooses the standard Euclidean Riemannian metric at the vector (0,0,0), then the group structure may be used to pull the metric back to a left invariant metric at each point. The result is this: ds^ = exp(2z) dx^ + exp(-2z) dy^ + dz^ at the point

(x,y,z).

Geometric group theory

289

3.3. The Cay ley graphs of the examples and the geometries with which they are associated One begins to understand both the sample groups and the geometries with which they are associated as one constructs the Cay ley graphs. The construction was left to the reader in Section 3.1; but now we describe the graphs for those who want to check their constructions or who failed to succeed in constructing them. Before we begin, we give an easy example of how the nature of the geometries allows one to prove a simple group theoretic statement: In reference to Gromov's polynomial growth theorem, we have observed the difference in growth rates between the free Abelian group Z^ on two generators and the free nonAbelian group Z * Z on two generators. We return to these simple examples. It is obvious the Z * Z does not embed in Z^ since the former is non-Abelian, the latter Abelian. But Z ^ Z does embed in all of the fundamental groups of surfaces having negative Euler characteristic. What about the fundamental group K of the Klein bottle which has 0 Euler characteristic? The group K is non-Abelian. Does it contain an embedded copy of Z * Z. As one of the simplest sample applications of geometry to group theory that we know, we use geometry to prove that Z :¥ Z does not embed in K. PROPOSITION. The fundamental group K of the Klein bottle does not contain an embedded copy of the free non-Abelian group Z ^Z. PROOF.

The group K has presentation K — [x,y \xyx~^y = l).

It is easy to see from the recognition theorem that the Cayley graph of K may be realized as the unit square lattice in the complex plane with all horizontal edges pointing to the right being labelled x, vertical edges pointing upward over even integer values being labelled y, vertical edges pointing downward over odd integer values being labelled y. Thus, as in the case ofZ^.K has exactly An elements of minimal length n for all integer lengths n > 0. If Z * Z could be embedded in K, then the images of the generators would have fixed lengths bounded by some integer N, Thus the embedding of Z * Z would induce an embedding of the elements of length ^ n in Z * Z into the elements of length ^ N n in K. But there are 1 + 4 + 3-4 + 3 ^ •4H h 3^^-^ • 4 of the former but only 1 + 4 + 2-4 + 3-4 + |-n-A^-4ofthe latter. The exponential sum of the former, however, clearly surpasses the quadratic sum of the latter for large n, a contradiction. The reader will have surmised that the group examples from Section 3.1 and the standard geometries of Section 3.2 have been presented in parallel so that the groups act on the corresponding geometries. We now describe the Cayley graphs. D 3.3.1. Euclidean groups. The Cayley graph of the free Abelian groups Z" can be realized as the unit lattice in Euclidean n-space R^. The group acts geometrically on that space by translation in the direction of the standard unit vectors.

290

J.W. Cannon

The Fibonacci groups are so named because of the nature of the relators. The relators can be used to reduce the generating set to two elements. One loses thereby much of the symmetry of the presentation. Conway's expressed goal in defining the groups was to create presentations for his students giving fairly simple groups in such a way that it was not at all obvious just how simple the groups were. In particular, Fs is the cyclic group of order 11, but the proof that this is so requires consideration of huge powers arising from repeated exponentiation as one moves cyclically through the defining relators. The group F6 acts geometrically on Euclidean 3-dimensional space. It is probably the only Euclidean group in the sequence. The group F7 has order less than 30. All of the groups Fn for n large are infinite. Some questions that have not been adequately dealt with are the following. Given natural infinite families of presentations, can one deal with most of the groups of the family in a uniform way? Are most of them infinite? Are most of them negatively curved (wordhyperbolic) in the sense of Gromov? (See Section 4.) 3.3.2. Spherical groups. Poincare's dodecahedral space is finitely covered by the Euclidean sphere S^. The dodecahedron lifts homeomorphically to S^ and the 120 different lifts tile S^ and are equivalent under the covering translations which are Euclidean rotations of S^. One can build the universal cover and recognize it topologically as S^ in a manner equivalent to Todd-Coxeter coset enumeration. One first calculates that three dodecahedra meet along each edge. One can view the patterns of three dodecahedra as defining relators for the fundamental group or as giving the necessary local conditions for the building of a covering space. One then begins with one dodecahedron and begins to build the universal cover by sewing new disjoint copies of the fundamental dodecahedron along free faces labelled according to opposite face identifications which defined the dodecahedral space. One satisfies the relators by making sure to put only three dodecahedra around each edge. If one does the construction systematically, say by forming at each stage the star of the previous stage, then one obtains successively larger balls whose boundary patterns are beautiful and easily calculated. At the final stage, a single final dodecahedron fills the hole and completes the sphere. It is very satisfying to carry out the construction and to see the sphere inevitably close up after the conjoining of 120 dodecahedra. The examples that we have given of groups that act geometrically on the sphere have been particularly chosen so as to be obviously geometrically related to a sphere of some dimension. However, every finite group acts geometrically on every sphere: simply take the trivial action. 3.3.3. Hyperbolic groups. If the Poincare dodecahedral space is modified just a tiny bit, the group becomes hyperbolic in the sense that it acts geometrically on hyperbolic 3-space; the Seifert-Weber space is in fact hyperbolic, not a finite group, but infinite. While the torus group Z^ and the Klein bottle group K are Euclidean, the fundamental groups of those surfaces that have negative Euler characteristic are hyperbolic. They present particularly nice examples of groups whose Cayley graphs, though infinite, can be quickly constructed algorithmically. As an example, we take the fundamental group of the closed, orientable surface of genus 2. In the standard way, we cut this surface along a bouquet of four loops so as to create a topological octagon with labelling boundary word a\b\a^

b^ a2^2^2^ ^2 •

Geometric group theory

b2

291

a. -^

a, V

b , ' \ •,f

/'», 2

Fig. 5. Presentation for the fundamental group of the surface of genus two.

The universal covering space of the surface is sewn together from infinitely many disjoint copies of this labelled octagon. We begin with one such octagon. It forms a disk with boundary label. We also attach labels of 1 to each of the vertices to indicate that, in this disk, there is only one octagon adjacent to this vertex at the present time. We now proceed by induction. We assume inductively that we have a disk with boundary label sewn together from finitely many disjoint copies of our original labelled octagon. We assume that, in addition to the boundary edge labels, there are labels of 1, 3, 5, or 7 attached to the vertices of this disk which indicate how many octagons are already adjacent to that vertex. We assume further that each of the labels of 3 or 5 or 7 are in fact separated by at least one 1 (in fact by at least five Ts). We form the next disk from this old disk by sewing new copies of the octagon along the boundary edges of our old disk, one disk along each edge except when we come to a vertex labelled with a 7. If a vertex is labelled with a 7, then, in order to form a covering space, a space locally homeomorphic to our original surface at that vertex, we are only allowed one more octagon. That final octagon must be sewn along the two adjacent edges which are joined at vertex labelled 7 to give eight octagons at that vertex. It is easy to see inductively that our new labelled disk satisfies the required inductive hypothesis. By a theorem entirely analogous to the recognition theorem, or by a suitable application of the recognition theorem to the fundamental group of the surface, it is easy to see that we construct larger and larger labelled disks whose infinite union is the universal cover of the surface. The reader should carry out several stages of the process in order to see just how compelling the construction is. The readers who have successfully constructed the universal cover of the surface in the previous paragraph might test their skills with the construction of the tilings of the hyperbolic plane and of hyperbolic 3-space associated with the reflection groups in the right-angled pentagon and the right-angled dodecahedron. These groups have fixed points, and so they do not define a 2-manifold or a 3-manifold in the way that a geometric action without fixed points would do. However, it is easy to write down finite presentations for these groups by applying a very important theorem of Poincare, called Poincare's theorem for fundamental polygons and polyhedra. A good reference to the theorem is [33]. Maskit's work in general gives a very accessible approach to the study of hyperbolic groups. See, for example, [34].

292

J.W. Cannon

1

1 Old disk boundary

New boundary with new added fundamental domains 1

1

1 1

iL-ii

1 1 >l-]i

New disk boundary

1

1

1

Fig. 6. Building the universal cover of the surface of genus two recursively. The four fundamental local patterns.

One should also see the work of Robert Riley [42] to see some rather fine applications of this theorem to 3-manifolds. In our setting, Poincare's theorem implies that one can construct something like a universal covering space by sewing together disjoint copies of the pentagon or dodecahedron in the freest possible fashion subject to the requirements that exactly four pentagons come together at each vertex and exactly four dodecahedra come together around each edge. In the pentagon case, one constructs labelled disks for which the vertices are all labelled 1 or 3, vertices labelled 3 separated by at least two vertices labelled 1. The construction is then exactly like the construction of the universal cover of the surface of genus 2, except that there are fewer local patterns to recognize. In the dodecahedron case, if one builds at each stage the complete star of the previous stage, then one finds recursively that every stage is a 3-cell and that the boundaries of the 3-cells are labelled in the following recursive fashion. Thus, although one cannot actually carry out the infinitely many steps required to form the entire infinite pattern, one can carry out as many of the finite stages of the construction as time and space permit, and the recursion required for going from one stage to the next is completely clear.

Geometric group theory

293

Fig. 7. Building the universal covering orbifold of the right-angled dodecahedral reflection group recursively. The three fundamental local boundary patterns.

3.3.4. The group defined in Section 3.1.4 is obviously associated with the very simple product geometry S^ x R. The soccerball factor is of course included only for aesthetic reasons. Since the factor 5^ is compact, it may be entirely ignored up to quasi-isometry. A group acts geometrically on S^ x R if and only if it acts geometrically on the line R itself. The groups that so act are determined by Stallings's theorem on the ends of groups. 3.3.5. To obtain a group that acts geometrically on H^ x R, one may take the product of a group that acts geometrically on H^ and multiply it by Z. The free non-Abelian group Z^Z does not act geometrically on H^. However it does act isometrically and properly discontinuously on ^ ^ in such a way that the quotient, while not being compact, has finite area. The fundamental group G of the trefoil knot complement is almost the product of Z * Z with Z. The commutator subgroup G' of G is the fundamental group of the incompressible, orientable Seifert surface of the knot, which is a torus with one hole, and which therefore has fundamental group Z^Z. The center C of the group G is isomorphic with Z. The internal product of G' and C in G is isomorphic with the external product of the same groups and has finite index in G. Thus, if we form the Cay ley graph of G, we will have a reasonable approximation to the geometry H^ x R. One of many ways to form the Cayley graph is as follows. In Euclidean jcyz-space, form an infinite homogeneous tree in the plane z = 0, three edges meeting at each vertex, one vertex at the origin. Label each vertex of the tree alternately as either up or down, with the origin being up. Label each edge of the tree as red, white, or blue in such a manner that at each vertex there are edges of each color.

294

J.W. Cannon

0

(arrow tip)

(arrow tail)

Fig. 8. One horizontal layer of the Cay ley graph of the fundamental group of the complement of the trefoil knot.

Now take a copy of this graph in each of the planes z = integer, but change the labelHng in the following way. As one moves up or down from one level to an adjacent one, change all up vertices to down vertices, all down vertices to up vertices. As one moves up from one level to an adjacent level, change red edges to blue edges, blue edges to white edges, white edges to red edges. Conversely, as one moves down from one level to an adjacent level, change red to white, white to blue, blue to red. Finally, join all of the levels together to form a single graph in the following way. Move all up vertices up 1/2 unit. Move all down vertices down 1/2 unit. Direct all edges so that they begin at an up vertex and end at a down vertex. We recommend that the reader use the recognition theorem to show that the graph just described is the Cayley graph of the trefoil knot group. Here are some things that are easy to read from the Cayley graph. (1) There is a natural homomorphism onto Z; indeed, just collapse the relator diagram so that every directed edge points vertically downward, or collapse the entire graph to the vertical straight line through the origin with each edge of the

Geometric group theory

295

graph going to a vertical interval of length 1. (2) The group is non-Abelian; simply follow paths labelled by two different generators in the two possible orders. (3) The element with labels red-white-blue-red-white-blue lies in the center of the group; indeed, check that it commutes with the three generators. (4) The commutator subgroup is the set of vertices lying in the plane z = 0. We note that other knot groups are associated, in general, with different geometries. 3.3.6. We wish to describe the Cayley graph for the group given in Section 3.1.6. S = {xuyuX2, y2J\t

= [x\,yi][x2, yi] central).

It is clearly closely associated with the surface group So = {x\,yuX2,y2

I [x\,yi][x2,y2] = l},

whose Cayley graph we described above in Section 3.3.3. To form the Cayley graph of S we proceed as follows. We assume that the Cayley graph FQ of So has already been constructed and naturally embedded in the plane z = 0. (The most natural plane in which to embed it would be the hyperbolic plane, though that is not critical for our purposes.) We first concentrate only on the vertices. We take copies of the vertices in each plane z = integer. We next take a tree T in FQ which contains every vertex of FQ. NOW to each edge path P in FQ which begins at the identity vertex we assign a height n(P) as follows. We consider the path PQ in T which begins at the terminal endpoint of P and ends at the identity vertex of FQ. The closed path P • PQ encloses n(P) of the disks in z = 0 bounded by the graph FQ, where a disk is counted k times if the path P • PQ has winding number k with respect to a point of the interior of the disk. The path P represents an element of the group 5, namely that vertex which lies in the plane z — n(P) above or below the terminal point of P which lies in the plane z = 0. If paths P and P' differ by one generating letter, say F^ = P • x, where x is either xi, yi, X2, ^2, or their inverses, then their vertices are to be joined by an edge labelled by the generator jc. Finally, each vertex is to be joined to the vertex directly above it by an edge labelled t. This completes the construction of the Cayley graph. This graph with each edge assigned the length of 1 approximates the geometry SL2. 3.3.7. We shall see that the nil group of Section 3.1.7 has Cayley graph which can be constructed much in the fashion of the SL2 group of Section 3.1.6 (see the previous Section 3.3.6) or alternatively much in the fashion of the solv group of Section 3.1.8 (see the next Section 3.3.8). We recall the description. Our nilgroup A/^ is a matrix group consisting of those 3 X 3-matrices of the form

Define the three matrices

296

J.W. Cannon

Easy calculations establish the following relationships:

(

1 a b\ 0 1 c ) ; 0 0 1/

AB = BA; CB = BC. The first of the relationships shows that the three elements A, B, and C generate the group. The last two relators show that B lies in the center of the group, and an easy calculation shows that in fact B generates the center. We claim that in fact this matrix group has as presentation (A, B, C I [A, C] = B, [A, B] = 1, [C, B] = l), where [X, Y] denotes the commutator XYX~^Y~K The group given by this presentation clearly can be mapped homomorphically into the matrix group since the matrix group satisfies the three relators. The map is onto by the first relationship above. The three relators can clearly be used to put any element into the form C^A^ B^, and these forms map one to one into the matrix group so that the homomorphism is injective. We may now copy the construction of the Cay ley graph from the previous section: we start with a surface group (a free-Abelian torus group Z ^ Z generated by A and C); we then delete the relator that says that A and C commute and replace their commutator instead by a new generator B (analogous to the generator t of the previous section). We then require that this new generator be central. There is an alternative way of constructing the Cayley graph which uses a little bit of linear algebra. We shall use an identical construction in the next Section 3.3.8. In order to form the Cayley graph, we first modify the presentation by the substitutions of generators a = A~\ p = B, and t = C. The presentation then takes the form {a,p,t I [Qf,^] = l, tar^

=ap, tpr^ = p).

We use the matrix

(1 ?)" Here is the construction: In each integral z-level put a copy of the unit lattice, the graph of the free Abelian group Z^ generated by a and p. Move the lattice in z-level n by the linear mapping

c :y

Geometric group theory

297

Thus, the further one moves away from the level z = 0, the further the lattice deviates from the square lattice. Since the matrix is invertible over the integers, the vertices of the different lattices are still the integer points. Hence we may join each vertex to the one above it by an edge labelled t. The Cay ley graph is now the union of the distorted integer lattices and the edges labelled t. Though the form of the construction is identical with that of the next section, the associated geometries are different because the monodromy matrix associated with conjugation by t is not hyperbolic but consists instead of a single Jordan block with multiple eigenvalues equal to 1. 3.3.8. Consider the group with the following presentation: {a,b,t\aba~^b~\

tat~^ =ab,

tbt~^=ab^).

In each integral z-level put a copy of the unit lattice, the graph of the free Abelian group Z^ generated by a and b. Move the lattice in z-level n by the linear mapping

(1 D" Thus, the further one moves away from the level z = 0, the further the lattice deviates from the square lattice. For n large and positive, every edge seems to point more and more in the eigendirection of the matrix associated with the eigenvalue greater than 1. For n large and negative, every edge seems to point more and more in the eigendirection of the matrix associated with the eigenvalue smaller than 1. However, since the matrix is invertible over the integers, the vertices of the different lattices are still the integer points. Hence we may join each vertex to the one above it by an edge labelled t. The Cay ley graph is now the union of the distorted integer lattices and the edges labelled t. 4. Abstract classes of groups The geometries and groups which we have explored in Section 3 have served as models for the development of geometric group theory over the last 20 years. The goal has been to develop combinatorial techniques to deal with all geometric 3-manifold groups and their generalizations. A great deal of success has been attained in at least three abstract classes of groups: (1) Gromov's word-hyperbolic groups which generalize the geometric properties of the groups which act geometrically on hyperbolic non-Euclidean geometry; (2) the automatic groups which distil the asymptotic recursive and computational properties noted by Cannon in his study of the same class of groups; and (3) the CAT(O) groups which attempt to reach out from groups of negative curvature to those of curvature ^ 0. In this section we give a short outline of these three subjects. In the past decade there have been a number of books and many, many papers written on these subjects. We cannot hope to duplicate them in this survey. In the midst of this development, there has been a vast explosion of knowledge about geometric invariants of groups. We shall discuss some of these invariants in Section 6.

298

J.W. Cannon

The whole development is somewhat reminiscent of the original development of combinatorial group theory originally following the early geometric work of Max Dehn. The early geometric insights became quite rapidly obscured as people turned to the more abstract and combinatorial realizations of those geometric ideas. That development led to small cancellation theory, which Gromov's work has once again made geometric rather than so exclusively combinatorial.

4.1. Word hyperbolic or negatively curved groups The fundamental paper is [29]. The paper is truly an essay in the sense that it simply outlines a vast amount of work and, when it gives details at all, they are often sketchy or even questionable. Nevertheless, this is a tremendously important paper. Gromov has connected group theory with global differential geometry, and in that aspect of the work he has been very careful. As a consequence he is able to translate geometric theorems into group theory with great freedom. In this translation he has not been as careful, and so has left much work to be completed. Many, many group theorists and geometers have therefore been involved in sorting out the details. This fundamental paper should not be read linearly since the ideas necessary to complete the arguments of one section may appear in a widely separated section. It is easier, but not as rewarding, to read one of the several books dedicated to making it all work precisely. Here are a few of the expository treatments [1,6,19,27]. Two closely related papers of Cannon [11,13] may be helpful as regards some of the results. Defining properties. Gromov isolated a whole family of equivalent properties of a metric space that are satisfied by manifolds having negative sectional curvature such as nonEuclidean hyperbolic space. Each is useful under appropriate circumstances, but the consensual favorite seems to be Rips's thin triangles property. We say that a proper path metric space X with path metric d has uniformly thin triangles (5) if, whenever JC j z is a geodesic triangle in X and whenever p is a point of one of the three sides, say p exy, then there is a point q in one of the other two sides, xz and yz, such that d{p, q) ^ 5. For example, any tree has uniformly thin triangles (0). What is not so obvious is that non-Euclidean hyperbolic space H^ has uniformly thin triangles (log(l -h V2)) = 0.88137 — As a consequence, geodesic polygons in such a space look essentially like slightly fattened trees. Thus geometric arguments become largely combinatorial in nature. A group is said to be word-hyperbolic if its Cay ley graph with respect to some finite generating set has uniformly thin triangles. A fundamental theorem shows that this property is independent of the finite generating set. Since the fundamental groups of closed manifolds with negative sectional curvatures are word-hyperbolic and since spaces that have uniformly thin triangles behave in the large just as negatively curved spaces behave in the large, the author also likes to call such spaces and groups negatively curved (in the large). Genericity. As odd as the thin triangles property seems at first glance, Gromov proved first of all that it is very easy to construct large numbers of spaces that satisfy this property. Among these spaces are the Cayley graphs of many groups. But even more, Gromov

Geometric group theory

299

claimed that the groups satisfying this property are in fact generic. A.Yu. Ol'shanskii [40] has confirmed that this is indeed the case: in a definite statistical sense, almost every finitely presented group is word-hyperbolic. The word, conjugacy, and isomorphism problem s. In the paper by Cannon [11] referenced above, Cannon proved that groups which act geometrically on hyperbolic space have word problem which is very efficiently solvable by a local shortening procedure known as Dehn's algorithm; Dehn developed this method in order to solve the word problem for surface groups. Cannon's proof works essentially without change for word-hyperbolic groups. The existence of such an algorithm for solving the word problem for a group is in fact equivalent to the word-hyperbolicity of the group. A suitable reference is [25] (see in particular the second appendix). Dehn also solved the conjugacy problem for surface groups. This problem asks for an algorithm which decides whether two words represent conjugate elements in the group. Cannon showed that Dehn's solution to that problem also extended to groups acting geometrically on hyperbolic space and the proof extends to word-hyperbolic groups. Dehn also gave an algorithm to determine when two surface groups are isomorphic. Zlil Sela has managed to solve the isomorphism problem for word-hyperbolic groups. Gromov has given other simple procedures to solve the word problem in negatively curved groups. Important unsolved problems. QUESTION.

The following problems seem to be very difficult.

Does every word-hyperbolic group have a torsion free subgroup of finite

indexl QUESTION. IS

every word-hyperbolic group residually finitel

One of the fundamental properties of a word-hyperbolic group is that it has a welldefined space at infinity. This space at infinity intervenes at many places in theorems about word-hyperbolic groups. QUESTION. Suppose a word-hyperbolic group has as its space at infinity the 2-sphere S^. Does it follow that the group acts geometrically on hyperbolic 3-spacel

The space at infinity. The points at infinity are equivalence classes of geodesic rays in the Cayley graph of the group. Two rays are equivalent if they remain asymptotically a finite distance apart. Two points at infinity are near to one another if representative geodesic rays remain close to one another for a long time. The space at infinity is compact, metrizable, finite-dimensional. It compactifies the Cayley graph in such a way that the group acts not only on the Cayley graph by left multiplication but also on the union of the graph with the space at infinity. It has just recently been proved by Brian Bowditch [7] and Gadde A. Swamp [46] that, if the space at infinity is connected, then it is also locally connected.

300

JM Cannon

Nadia Benakli [4] has given examples of word-hyperbolic groups having the Sierpinski curve and the Menger curve as their respective spaces at infinity. Here is an unresolved question. QUESTION. Characterize topologically those topological spaces that can occur as the space at infinity for a negatively curved group.

4.2. Automatic groups The author learned about Cayley graphs and Todd-Coxeter coset enumeration from the book by Coxeter and Moser [17]. Taking it as an extended exercise to construct the Cayley graph of all of the groups whose presentations were given in the book, the author discovered many beautiful graphs. One of the first was the graph of the trefoil knot group described in Section 3. Since the author had somehow been led to expect that it would be essentially impossible to understand the graphs of all except the simplest infinite groups, he was surprised to find again and again that the asymptotic structure of the Cayley graphs he considered were eminently understandable. This discovery led to the paper cited above (Cannon [11]). Particularly Theorem 7 of that paper considered the combinatorial pictures as one looked toward infinity from the identity vertex through different vertices of the group. That theorem showed that there were in fact, for negatively curved groups, only finitely many such pictures. Once one had discovered these finitely many pictures, one knew what was going to happen forever in the Cayley graph. As a consequence, one learned that the language of geodesic words in a negatively curved groups form what is called a regular language. Thurston was later to distil the essence of this situation to create automatic group theory. The basic reference for automatic group theory is [22]. We shall give a short introduction to automatic group theory here. Finite state automata. We start with a finite alphabet A. We consider the set W of (finite) words that can be written in the alphabet A, including the empty word. We now consider a finite directed and labelled graph A with the following properties. (1) From each vertex emanates one and only one edge labelled by each letter of the alphabet A. Consequently, beginning at any given vertex, and given any word w eW, there is one and only one path labelled with the word w. It leads from the given vertex to another vertex of the graph A. (2) One vertex of the graph A is designated as the start vertex UQ of A. (3) Some family S of vertices of A (possibly the empty subset) is designated as the family of success vertices or accept vertices of A. The graph A with its start vertex and its accept vertices is called difinitestate automaton. It defines a subset of W called its language L{A) of accepted words in the following way. Take a word w ^W. Take the unique path which begins at the start vertex VQ and is labelled

Geometric group theory

301

by the word w. The word is accepted if the terminal vertex of the path is an accept or success vertex. The set of accepted words is the language L{A). A language is called regular if it is the language accepted by some finite state automaton. The set of regular languages has very nice set theoretic properties. It is closed under unions, intersections, complements, and various other nice logical operations. Such languages lend themselves very nicely to machine computation. The search operations in most word processing units are based on the theory of regular expressions or regular languages. This fact motivates the name of the reference, "Word processing in groups." The fellow traveller property. It is convenient to parametrize an edge path in a Cayley graph by arc length. It is also convenient to pad such a finite path so that the path is parametrized by [0, oo); one simply has the path sit forever at the final vertex after it reaches that vertex. With those conveniences seen to, we can define the k-fellow traveller property. We say that two paths satisfy the /:-fellow traveller property if, for each parameter t e [0, oo), the image points in the two paths are within k of one another in the Cayley graph. Definition of automatic group. A group G is said to be automatic if there is a finite alphabet A, a map 0 : A -^ G, and a finite state automaton A on the alphabet A, and a positive number k having the following properties. (1) First we note that the map 0 can be extended to the entire set W of words written in the alphabet A by concatenation. The extended map c/y.A^^G must map the regular language L(A) c W onto G. That is, the elements of G must all be represented by accepted words of A. (2) If u and v are accepted words of G and if the associated paths in the Cayley graph, which begin at the identity vertex and are labelled respectively by u and i;, end within a distance 1 of each other, then these paths satisfy the /:-fellow traveller property. There are two key facts about automatic groups. The first key fact is that in such groups one can perform computations very efficiently. David Epstein, Derek Holt, and Sarah Rees at Warwick University have developed an entire suite of computer programs designed to discover automatic structures on finitely presented groups when they exist and to create all of the finite state automata necessary to make quick and efficient calculations. Their suite of programs is called "aut," for automatic groups. The second key fact is that such groups are generic in the class of all groups. In fact, the fundamental example from automatic group theory is that every word hyperbolic group is automatic. As the regular language required, one may take the language of geodesic words in the generating set of the group. Property (2) is then a reasonably easy consequence of the thin triangles property: geodesies in a negatively curved group which have their endpoints near to one another must stay at all times near to one another. Using the set theoretic properties of regular languages, one can clean up the automata used. In particular, one can pass to an accepted language of unique representatives for the group G. That is, one can represent the elements of G by quickly calculable normal forms.

302

J.W. Cannon

4.3. CAT(O) groups As we noted in Section 4.1, negatively curved or word-hyperbolic groups form a generic class of finitely presented groups. But of course all 3-manifold groups are interesting to topologists, and the negatively curved groups certainly do not include the fundamental groups associated with five of the other standard 3-dimensional geometries (the groups associated with S^ being finite are negatively curved in a trivial sense; the groups associated with S^ X R are negatively curved since Z as a rather trivial tree is negatively curved). If geometry is to be used as a guiding principle in the study of groups, then one has to seek generalizations which pull in other geometries. Various generalizations have been considered. All of these have included in addition at least the Euclidean groups. The generalization to automatic groups introduced in the previous section is computationally based. The generalization of this section is geometrically based. Among the fundamental references are [2,9,10]. Let X denote a geometry. If xyz is a geodesic triangle in X, then the sides satisfy the triangle inequality so that there is a comparison triangle x'y'i! in the Euclidean plane which has identical edge lengths: \xy\ = Wy'\, \yz\ = \y'z^\, and |zjc| = \z^x^\. There is an obvious correspondence between the two triangles which is an isometry on corresponding edges. We say that Z is a CAT(O) space if, for each choice of vertex v of xyz and for each choice of point w on the edge opposite i;, d(v, w) ^ d(v\ w') where v' and w^ are the points of the comparison triangle x'y'z^ corresponding to the points v and w of xyz. That is, in the space X triangles are at least as thin as the triangles in Euclidean space. The CAT(O) property is intended to generalize the notion of space with sectional curvatures ^ 0. DEFINITION.

DEFINITION.

We say that a group is a CAT(O) group if it acts geometrically on a CAT(O)

space. An interesting first example of a CAT(O) space that is neither negatively curved nor Euclidean is the Cartesian product of the Cayley graph of the free non-Abelian group on two generators and the Cayley graph of the integers. The first is an infinite tree in which four edges come together at each vertex. The second is the line. The product group Z ^ Z X Z acts geometrically on this CAT(O) space and is therefore a CAT(O) group. The class of CAT(O) spaces shares many of the important characteristics of negatively curved spaces. In particular such spaces can be locally recognized so that there are nice constructions which stay in this class of spaces. Also the fundamental decision problems, the word and conjugacy problems, can be solved in this class of spaces, though their solutions are not as nice as they are for negatively curved groups. CAT(O) spaces have a natural "visual" space at infinity which can serve as space at infinity for CAT(O) groups; this visual space at infinity has as its set of points the geodesic rays emanating from some base point; two rays are close together if they remain close together over a large initial segment of each. Unfortunately, these spaces at infinity are not nicely attached to the groups in question. For example, with the CAT(O) group Z^Z xZ described in the previous paragraph.

Geometric group theory

303

the space at infinity is the suspension of the Cantor set, and the group does not act continuously on the union of the Cayley graph and the space at infinity. The appropriate reference to this fact is [8]. 5. Other geometries to explore We have considered primarily the standard 2- and 3-dimensional geometries arising in the study of topological manifolds. However, there are many other geometries which might be studied as sources for appropriate geometric group theory. There are, of course, all of the classical Lie groups. One particularly relevant reference is [49]. One might study the affine geometries, the projective geometries, the conformal geometries. One might study the groups associated with important classes of Riemannian manifolds, such as the Kahler manifolds or Einstein manifolds. In order to have a good theory, as pointed out by Gromov and Bridson, it is particularly important to be able to construct examples. It is often amazingly difficult to construct examples. One should mention in passing that there are two essentially nongeometric sources of examples for combinatorial or geometric group theory, namely logic and algebra. Much work remains to be done in connecting these sources with geometry. A particularly interesting recent paper in this regard is the paper of Nabutovsky [39]. Nabutovsky gives "a lower bound for the number of contractible closed geodesies of length ^ jc on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for m (M), thus answering a question posed by Gromov." 6. Other geometric invariants We mention here just a few of the geometric invariants that have been introduced and studied by geometric group theorists in the past decade. We apologize in advance for the many that we omit. We refer to possible entry papers into the topics cited rather than to the most recent or most complex papers on the subject. The most amazing collections of geometric invariants appear in the two papers of Gromov [29,30]. One of the very interesting invariants introduced by Gromov is that of isoperimetric inequalities: If u; is a word representing the identity element in G, then w can be expressed as a product of conjugates of defining relators; this product may be thought of as defining a disk bounded by w and tiled by small subdiscs corresponding to the relators of the product; the number of small disks required is called the area of the disk. The isoperimetric problem asks for the smallest area of a disk bounded by u;. A Dehn function f(n) for the group is a function such that, if K; is a relator of length ^ n, then w bounds a disk of area < / ( « ) . Gromov has characterized the word-hyperbolic or negatively curved groups as those which admit a linear Dehn function. Great progress has been made in calculating the Dehn functions and the possible Dehn functions of groups. Very interesting results about isoperimetric functions on groups can be obtained by applying homological and cohomological methods. We mention two references [3,24].

304

J.W. Cannon

Our personal attempt to generalize negatively curved groups so as to include the Euclidean groups involved the notion of almost convex groups. Here are two references [12,15]. Casson and Poenaru have suggested a generalization which has application to the following problem: when is the universal covering space of an irreducible, orientable 3-manifold homeomorphic with Euclidean 3-dimensional space R^l The required condition C2 is a property of the fundamental group of the manifold and is called an isodiametric inequality. It may be interpreted as a measure of how efficiently the Todd-Coxeter enumeration completes the correct construction of the ball of radius n in the Cay ley graph of that group. If the process works efficiently enough, then the universal cover is, in fact Euclidean. An exposition of this work appears in the reference [26]. Much fruitful work has been expended in extending Stallings's work on the ends of groups. One very beautiful consequence is the accessibility theorem for finitely presented groups proved by Dunwoody [21]. Dunwoody proves that, in a finitely presented group, it is impossible to factor it nontrivially an infinite number of times as a free product with amalgamation or HNN extension. Others have attempted to analyze the topology of the group near the ends of the group. Geoghegan and Mihalik have studied what they call semistability of the group at an end of the group. They have also studied iht fundamental group at infinity, and other topological properties of the ends of a group. We give only two entry references [23,36].

References [1] J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro and H. Short, Notes on word-hyperbolic groups. Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger and A. Verjovsky, eds, World Scientific, Singapore (1991), 3-63. [2] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhauser, Boston (1985). [3] G. Baumslag, C.F. Miller III and H. Short, Isoperimetric inequalities and the homology of groups. Invent. Math. 113(1993), 531-560. [4] N. Benakli, Polygonal complexes. I. Combinatorial and geometric properties, J. Pure Appl. Algebra 97 (1994), 247-263. [5] R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext, Springer, BerUn (1992). [6] B.H. Bowditch, Notes on Gromov's hyperbolicity criterion for path-metric spaces. Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger and A. Verjovsky, eds. World Scientific, Singapore (1991), 64-167. [7] B.H. Bowditch, Connectedness properties of limit sets. Trans. Amer. Math. Soc. 351 (1999), 3673-3686. [8] PL. Bowers and K. Ruane, Boundaries of nonpositively curved groups of the form G x Z'^, Glasgow Math. J. 38 (1996), 177-189. [9] M.R. Bridson, Geodesies and curvature in metric simplicial complexes. Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger and A. Verjovsky, eds. World Scientific, Singapore (1991), 373^63. [10] M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin (1999). [11] J.W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. [12] J.W. Cannon, Almost convex groups, Geom. Dedicata 22 (1987), 197-210. [13] J.W, Cannon, The theory of negatively curved spaces and groups, Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, T. Bedford, M. Keane and C. Series, eds, Oxford Univ. Press, Oxford (1991), 315-369. [14] J.W. Cannon, Colored graphs. Expository preprint, 27 pages. [15] J.W. Cannon, W.J. Floyd, M.A. Grayson and W.P Thurston, Solvgroups are not almost convex, Geom. Dedicata 31 (1989), 291-300.

Geometric group theory

305

[16] J.W. Cannon, W.J. Floyd, R. Kenyon and W.R. Parry, Hyperbolic geometry. Flavors of Geometry, S. Levy, ed., MSRI Publ., Vol. 31, Springer, Berlin (1997), 59-115. [17] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik, Bd. 14, 2nd ed., Springer, Berlin (1965). [18] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik, Bd. 14, Springer, Berlin (1972). [19] M. Coomaert, T. Delzant and A. Papadopoulos, Notes sur les groupes hyperboliques de Gromov, Lecture Notes in Math., Vol. 1441, Springer, Berlin (1990). [20] M. Dehn, Uber unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1912), 116-144. [21] M.J. Dunwoody, The accessibility of finitely presented groups. Invent. Math. 81 (1985), 449-457. [22] D.B.A. Epstein, with J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson and W.R Thurston, Word Processing in Groups, Jones and Bartlett, Boston (1992). [23] R. Geoghegan and M. Mihalik, The fundamental group at infinity. Topology 35 (1996), 655-669. [24] S.M. Gersten, Isoperimetric functions of groups and exotic cohomology. Combinatorial and Geometric Group Theory (Edinburgh, 1993), London Math. Soc. Lecture Note Series, Vol. 204, Cambridge Univ. Press, Cambridge (1995), 87-104. [25] S.M, Gersten and H.B. Short, Small cancellation theory and automatic groups. Invent. Math. 102 (1990), 305-334. [26] S.M. Gersten and J.R. Stallings, Casson's idea about 3-manifolds whose universal cover is R^, Intemat. J. Algebra Comput. 1 (1991), 395-406. [27] E. Ghys and P. de la Harpe, eds, Sur les groupes hyperboliques d'apres Mikhael Gromov, Progr. Math., Vol. 83, Birkhauser (1990). [28] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53-73. [29] M. Gromov, Hyperbolic groups. Essays in Group Theory, S.M. Gersten, ed., MSRI Publ., Vol. 8, Springer, Berlin (1987), 75-265. [30] M. Gromov, Asymptotic invariants of infinite groups. Geometric Group Theory: Proceedings of the Symposium Held in Sussex, 1991, Cambridge Univ. Press, Cambridge (1993). [31] B. Iverson, Hyperbolic Geometry, London Math. Soc. Student Texts, Vol. 25, Cambridge Univ. Press, Cambridge (1993). [32] R.C. Lyndon and RE. Schupp, Combinatorial Group Theory, Springer, Berlin (1977). [33] B. Maskit, On Poincare's theorem for fundamental polygons. Adv. in Math. 7 (1971), 219-230. [34] B. Maskit, Kleinian Groups, Springer, Berlin (1988). [35] W.S. Massey, Algebraic Topology: An Introduction, Springer, New York (1967). [36] M. Mihalik, Semistability at oo, oo-ended groups and group cohomology. Trans. Amer. Math. Soc. 303 (1987), 479^85. [37] J.W. Morgan and H. Bass, The Smith Conjecture, Academic Press, Orlando (1984). [38] G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Stud., Vol. 78, Princeton Univ. Press, Princeton, NJ (1973), 195. [39] A. Nabutovsky, Fundamental group and contractible closed geodesies, Comm. Pure Appl. Math. 49 (1996), 1257-1270. [40] A.Yu. Ol'shanskii, Almost every group is hyperbolic, Intemat. J. Algebra Comput. 2 (1992), 1-17. [41] J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Math., Vol. 149, Springer, New York (1994). [42] R. Riley, Applications of a computer implementation of Poincare's theorem on fundamental polyhedra. Math. Comp. 40 (1983), 607-632. [43] J.J. Rotman, The Theory of Groups, AUyn and Bacon, Boston (1973). [44] P Scott, The geometries of 3>-manifolds. Bull. London Math. Soc. 15 (1983), 401^87. [45] J.R. Stallings, On torsion-free groups with infinitely many ends, Ann. Math. 88 (1968), 312-334. [46] G.A. Swamp, On the cut point conjecture. Electron. Res. Announc. Amer. Math. Soc. 2 (2) (1996), 98-100. [47] W. Thurston, The geometry and topology of 3-manifolds, Lecture Notes, Dept. of Math., Princeton Univ. Press, Princeton, NJ (1978). [48] WP. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, S. Levy, ed., Princeton Univ. Press, Princeton (1997). [49] J. A. Wolf, Spaces of Constant Curvature, Publish or Perish, Inc., Berkeley, CA (1977).

This Page Intentionally Left Blank

CHAPTER 7

Infinite Dimensional Topology and Shape Theory Alex Chigogidze* Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada E-mail: chigogid@ math, usask. ca

Contents 1. Introduction 2. "Infinite dimensional topology" modulo a complex 2.1. Extension types of complexes 2.2. Extension dimension 2.3. Absolute extensors modulo a complex 2.4. Z-sets 2.5. Strongly [L]-universal spaces 2.6. [L]-homotopy 2.7. [L]-shape 2.8. Open problems 3. The case L = {pt}: "genuine" infinite dimensional topology 3.1. Examples of absolute extensors 3.2. Topology of/^-manifolds 3.3. Topology of/?^-manifolds 3.4. Topology of r-manifolds 3.5. Shape theory 4. The case L = 5": "n-dimensional" infinite dimensional topology 4.1. Menger manifolds 4.2. NobeUng manifolds 4.3. n-shape theory 5. Non-metrizable manifolds 5.1. r-limitation topology 5.2. /^-manifolds 5.3. /?^-manifolds 6. Topological groups 6.1. Abelian case - the Whitehead problem and characterization of Torus groups 6.2. Non-abelian case References

*The author was partially supported by NSERC research grant. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

307

309 310 310 312 314 317 322 327 333 335 339 340 343 346 349 351 352 352 355 357 357 360 361 363 364 365 366 366

This Page Intentionally Left Blank

Infinite dimensional topology and shape theory

309

1. Introduction Infinite dimensional topology, in its traditional understanding, is a branch of geometric topology which studies infinite dimensional spaces, arising naturally in topology and functional analysis. Objects such as the Hilbert cube Q (i.e., the countable infinite product I^ of the closed interval), the Hilbert space I2 (topologically the countable infinite product R^ of the real line), a-compact locally convex Hnear spaces U and a, as well as manifolds modeled on them, are typical examples of spaces attracting attention of experts in this field. Although the first nontrivial results in this direction (for example, the theorem of Keller [109] on the topological homogeneity of / ^ ) appeared more than sixty years ago, systematic studies of infinite dimensional manifolds became possible only after discovery by K. Borsuk of ANE-spaces. Standard tools of the theory of ANE-spaces have their roots in classical methods of piecewise linear topology as well as in homological and homotopical methods of CW complexes. Long awaited topological characterizations of the above mentioned model spaces, as well as of the corresponding manifolds, were obtained by Toruiiczyk [140,141] in terms of certain strong universality conditions (see also [21,22, 89,120]). The next major step in infinite dimensional topology, as it turned out later, was made within finite dimensional topology. In his fundamental thesis [20] M. Bestvina characterized the (finite dimensional!) universal Menger compactum /x" and developed the theory of /x"-manifolds. It is crucial to emphasize that the characterization in this case was given within the theory of ANE(/i)-spaces in terms of the n-dimensional version of the above mentioned strong universality property. An important philosophical consequence of Torunczyk's and Bestvina's works can roughly be expressed as follows: the n-dimensional universal Menger compactum /x^ and not the standard n-dimensional cube /" should be considered as a true «-dimensional counterpart of the Hilbert cube I^; these two spaces cannot be distinguished by means of "w-dimensional tests". There even is a profound analogy between the /'^-manifold and /x'^-manifold theories. Almost every statement from one of these theories has its counterpart in another. The same applies to the theories of shape and «-shape. Several examples of such pairs of statements can be found in Sections 3 and 4. These results led, in turn, to a development of the theory of Nobeling manifolds (shortly, v'^-manifolds) [51] as "finite dimensional" analogs of /?^-manifolds. Is it really necessary to present all particular results of each of the mentioned theories separately in order to have a nonzero probability that a nonexperienced reader, specifically a beginner in the field, understands basic ideas and tastes a flavor of typical techniques in the subject? Is there a general point of view incorporating both infinite and finite dimensional theories? Below the reader will find an attempt at such an approach. We present basics of an "(in)finite dimensional topology modulo a CW complex", define major concepts and provide necessary constructions. Some of the statements are accompanied by proofs (or their sketches). For a given complex L (actually for a class [L] of complexes associated with L) we develop a theory of ANE([L])-spaces and introduce the corresponding homotopy machinery based on the relation of [L]-homotopy. Following the classical guidehnes (known

310

A. Chigogidze

as "Edwards' strategy") we discuss the problem of existence of model [L]-dimensional strongly [L]-universal ANE([L])-spaces. Generally speaking, for each L, one expects to have two model spaces: /x^^^ - the Menger compactum modulo L and v^^^ - the Nobeling space modulo L. For which L do these objects exist? It turns out that noncompact model spaces, the Nobeling spaces modulo L, can be constructed for each finitely dominated complex. Whereas the Menger compacta modulo L are constructed only for finite L. Such objects do not always exist as shown by the case L = K(Z,n). An important problem whether compacta /x^^^ can be constructed for finitely dominated L also remains open. Two particular complexes, namely the point {pt} and the n-dimensional sphere 5", produce the theories of ANE- and ANE(n)-spaces, respectively. We will see that the [{pt}]homotopy coincides with the usual homotopy, whereas the [^"l-homotopy is precisely the n-homotopy in the sense of J.H.C. Whitehead [151] (see also [95]). The universal model spaces in these two particular cases produce the basic objects mentioned above: / ^ = /x^P^^, R^ = yM, /x" = fji^^ ^ and v" =v^^ \ After such a general discussion we return to a traditional form of presentation of classical topics of "genuine" infinite dimensional topology and shape theory. A small hope remains that a so organized overview of the subject will be helpful at least to some of the readers. Regrettably several important topics (such as infinite dimensional spaces arising in classical dimension theory, geometry of Banach spaces, classical M-manifolds, algebraic aspects of shape theory, appHcations of A^-theory etc.) are left without discussion. I would like to express my sincere thanks to the editors of this handbook. I am especially grateful to Dick Sher, who has carefully read the entire manuscript and has made many valuable comments and suggestions.

2. ^'Infinite dimensional topology" modulo a complex All spaces, with the exception of Section 5, are metrizable and separable. Maps are continuous. We call separable completely metrizable spaces the Polish spaces. All complexes considered in this paper are (unless it is explicitly indicated otherwise) countable and locally finite CW complexes. By V we denote the wedge operation, which unless otherwise specified is taken with arbitrary base points.

2.1. Extension types of complexes For spaces X and L, the notation L G A(N)E(X) means that every map f :A^^ L, defined on a closed subspace A of X, admits an extension f :X ^^ L (respectively, f .G ^^ L) over X (respectively, over a neighborhood G of A in X). Next we introduce a relation ^ for complexes. Following [76], we say that L ^ A' if for each space X the condition L G A E ( X ) imphes the condition K G A E ( X ) . Equivalence classes of complexes with respect to this relation are called extension types. The above defined relation < creates a partial order in the class of extension types. This partial order will still be denoted by ^ and extension type with representative L will be denoted by [L]. Note that under these definitions the class of all extension types has both maximal and minimal

Infinite dimensional topology and shape theory

311

elements. The minimal element is the extension type of the 0-dimensional sphere S^ (i.e., the two-point discrete space) and the maximal element is obviously the extension type of the one-point space {pt} (or, equivalently, of any contractible complex). It is important to understand the above introduced partial order on the set of all extension types of complexes. We present some examples. EXAMPLE 2.1.

(i) Clearly, 5" e AE(X) <^ dimX < n. (ii) Similarly, K(G, n) e AE(X) ^ dime X ^n. Here dime X stands for the cohomological dimension^ of X with coefficients in an abelian group G and K{G, n) denotes the corresponding Eilenberg-MacLane complex, i.e., a complex satisfying the following conditions: 7tn(K(G, n)) = G and 7tk(K(G, n)) = 0 for each (iii) Obviously [5"^] ^ [K(Z,n)]Mt[S''] (for n = 3) and [87] (for n = 2). (iv) Clearly

/ [/^(Z,w)]. The last part follows from [74]

[50]<[5i]<..-<[5"]<[r+^]<... and [pt] = sup{[5"]:« = l , 2 , . . . } . (v) Similarly [K(Z, 1)] < [K(Z, 2)] < . • • < [K(Z, n)] <

[K(Z,

n + 1)] < • • •

and [pt] = sup{[K(Z,n)]:

« = 1,2,...}.

(vi) min{[Ll[K]} = [Lv K]. (vii) [S^] = [L]<^ L is not connected, (viii) [S^] ^[L]^ L is connected. EXAMPLE 2.2.

(i) [L] < [Z(L)l Moreover, if L G A E ( X ) , then ZiL) e AE(X x [0,1]) (see [75]). (ii) L e AE(Z) =^Z(L)e AE(i:(Z)). EXAMPLE

2.3. Since [S^] ^ [L] for any complex L, it follows from Example 2.2 that^

[pt] = sup{[i:"(L)]:n = l , 2 , . . . } . Compare with Example 2.1(iv). See [83] for a comprehensive discussion of the theory of cohomological dimension. ^ U^ (L) denotes the iterated suspension of L.

312

A. Chigogidze

EXAMPLE 2.4.

(i) It follows from Example 2.1(vii), (viii) that there is no complex L such that [S^] < [L]<[S^l

(ii) There exists a 2-dimensional compactum X such that dime X = 1 for some group G. This implies that [S^] < [S^ v K(G, 1)] < [S^ (iii) Let L = M(Z2, n + 1) V S'''^^, where M(l2, n + 1) is the Moore complex. Obviously, [5^^] < [L] < [5^^+^. (iv) If [L] ^ [S% then [L] = [L^^^ V 5"]. EXAMPLE 2.5. It follows from [79] that the extension type [RP^] of the projective plane is not comparable with [S"^] for any n^2.

The Homotopy Extension Theorem implies the following trivial observation. PROPOSITION

2.1. IfL and K are homotopy equivalent complexes, then [L] = [K\

Observe that [5" v 5""^^] = [5"]. This shows that homotopy inequivalent complexes might have the same extension type. The following result shows that if only compact spaces are used in the definition of the relation ^ (see the beginning of Section 2.1), then every (not necessarily countable) complex is equivalent to a wedge of countable complexes. PROPOSITION

2.2 ([77]). Let Lhea complex, then

[L] =\\J

{K\ K is a countable complex such that

L^K]\.

2.2. Extension dimension The next step of our program is to define the extension dimension, ed(X), of a space X. DEFINITION 2.1 ([53,76]). The extension dimension of a space X is the minimum of extension types of complexes L satisfying the relation L € AE(Z):

ed(X)=min{[L]:

LGAE(X)}.

Of course extension dimension can be defined for larger classes of spaces. It was shown in [53] that using a spectral technique [51] all general questions can be reduced to similar ones for Polish spaces. 2.3 ([53]). For a given complex L there is a Polish space X^^^ such that ed(X) = [L]. If, in addition, L is finitely dominated, we may assume that X^^^ is compact. PROPOSITION

Generally speaking the extension dimension of a Polish space is not always defined (it is, if we consider arbitrary, not necessarily countable, complexes [76]).

Infinite dimensional topology and shape theory

313

The following question^ remains open. 2.1. Is the extension dimension of a metrizable compactum (or, more generally, of a Polish space) generated by a countable complex?

PROBLEM

Several standard statements of the classical dimension theory remain true for the extension dimension. PROPOSITION

2.4 ([53,76,145]). IfY c X, then ed(7) ^ ed(X).

PROPOSITION 2.5 ([53,76,145]). IfX = \J{Xk: k e co}, where Xk is a closed subspace ofX such that td{Xk) < [L], keo), then ed(X) ^ [L]. 2.6 ([53,123]). Let X = limS, where S = {Xk, /7^^^ ft;} be an inverse sequence consisting of Polish spaces such that Qd{Xk) ^ [L] for each k e co. Then ed(X) ^ [LI PROPOSITION

2.7. Let K be a locally compact polyhedron. (a) If dim K = n, then cd(K) = [S""]. (b) IfK is infinite dimensional, then ed(A^) = [pt].

PROPOSITION

PROOF. First we show that ed(/") = [5^"]. Clearly ed(/'') ^[S""]. Let L be a complex such that L G AE(/'^). By Propositions 2.4 and 2.5, L e AE(K) for each at most ^-dimensional compact polyhedron. Further, if Z is an at most «-dimensional metrizable compactum, then, by Freudental's theorem, Z is the limit space of an inverse sequence {Ki, p^-^ } consisting of at most «-dimensional compact polyhedra. By the above remark and by Proposition 2.6, L G A E ( Z ) . In particular, L G AE(/X"), where /x" denotes the n-dimensional universal Menger compactum. Now, if Y is an at most «-dimensional space, we may assume that Y C. /JL^. Consequently, by Proposition 2.4, L e AE(7). In other words S" G AE(X) =^ L e AE(X) for each space X. According to the definition of the above partial order, this means precisely that [5^] ^ [L] for each L satisfying the condition L G AE(/"). Therefore ed(/") = [5^]. This immediately implies that Qd{K) = [5"] for each n-dimensional locally compact polyhedron. Now assume that K is an infinite dimensional locally compact polyhedron and L is a complex such that L G AE{K). Since every contractible complex generates the extension type of the one-point space {pt}, it suffices to prove that L is contractible. Assuming that this is not the case, L must have at least one nontrivial homotopy group, say 7r„ (L) ^ 0. Let / : 5" —> L be a map generating a nonzero element of 7r„(L). Clearly K contains the {n H1)-dimensional disk B^^^ and since L G A E ( A ' ) we conclude that the map / : dB^^^ = S^ ^^ L has an extension over B^^^. This contradicts the nontriviality of the homotopy class [/] G 7r« (L) and finishes the proof. D

In the light of the previous statement the following result appears to be somewhat unexpected. ^ Compare with Proposition 2.2

314

A. Chigogidze

PROPOSITION 2.8 ([56]). For an arbitrary complex Lfor which there exists a metrizable compactum X such that ed(X) = [L], assuming Jensen's principle, there exists a differentiable, countably compact, perfectly normal and hereditarily separable A-manifold M^ such that ed(M^) = [L].

2.3. Absolute extensors modulo a complex In this subsection we introduce the notion of absolute extensors modulo L. DEFINITION 2.2. We say that a space X is an absolute (neighborhood) extensor modulo L, or shortly that X is an A(N)E([L])-space (notation: X e A(N)E([L])) if X G A(N)E(y) for each space Y with ed(y) ^ [L]. REMARK

2.1. It follows from the above definitions that L e AE([L]).

Example 2.1(vii), (viii) explains the following statement. PROPOSITION 2.9. IfL is not connected, then the class of A(H)E{[L])-spaces coincides with the class of K(^)E{{S^Y)-spaces. If L is connected, then every K(N)E{[L\)-space is an k{W^E{[S^Y^-space.

2.10. If{L] ^ [P], then the class of A(K)E{{P])-spaces is contained in the class of K(H)^({L\)-spaces. PROPOSITION

PROPOSITION

2.11. An open subspace of an ANE{[LY)-space is an ANE([L])-5'/7ac^.

PROPOSITION

2.12. A retract of an A(N)E([L])-^/7flc^ is an A(^)^{[LY^-space.

2.6. Observe that: (i) ANE([{pt}])-spaces are precisely ANE-spaces. (ii) ANE([5"])-spaces are precisely ANE(«)-spaces, i.e., LC"~^-spaces.

EXAMPLE

Next we introduce the concepts of [L]-soft and [L]-invertible maps [53]. DEFINITION 2.3. A map / : Z ^ 7 is said to be [Lysoft, if for each space B with ed(B) < [L], for each closed subset A of it, and for any two maps g.A^^X and h:B -^ Y such that fg = h/A, there is a map k.B^-X satisfying the conditions k/A = g and fk = h. If the above condition is satisfied in the cases when B = 0 we say that / is [L]-invertible.

Clearly, every [L]-soft map is [L]-invertible. [L]-invertible maps are surjective, while [L]-soft maps are open [53, Remark 5.10]. We Hst the following immediate corollaries. PROPOSITION

f:X-> (i)

2.13.

Y: XGA(N)E([L]).

(ii) yGA(N)E([L]).

The following conditions are equivalent for an {L\-soft map

Infinite dimensional topology and shape theory

315

In particular, X e AE([L]) if and only if the constant map X -> {pt} is [L]-soft. The following statement is important for our considerations. 2.1 (Proposition 5.9 [53]). Let Lbea complex and Xbea {Polish) space. Then there exists an [L]-soft map fix'-^x -^ X, where Yx is a {Polish) space with ed(yx) = [^]If, in addition, A is a closed subset ofX such that ed(A) < [L], then we may assume that fx/fx (^) • fx (^) -^ Ais a homeomorphism. THEOREM

COROLLARY 2.1 ([53,123]). For a given complex L there exists a Polish AE([L])space X^^^ such that Qd{X^^^) = [L] and X^^^ contains a {closed) topological copy of any {Polish) space Y with Qd{Y) ^ [L]. PROOF. Let / : X^^^ -^ / ^ be an [L]-soft map constructed in Theorem 2.1. Since / ^ is an AE([{pt}])-space, by Proposition 2.10, I^ is an AE([L])-space. Then, by Proposition 2.13, X^^^ is an AE([L])-space. Now let F be a space such that ed(7) ^ [L]. We may assume that 7 c / ^ . Since / is [L]-soft the inclusion map F ^^ / ^ admits a lifting iy'.Y ^ X^^^. Clearly iy is an embedding. D COROLLARY

2.2 ([53,122]). Every space X has a completion X such that ed{X) =

ed{X). There are spaces without cohomological dimension preserving compactifications [86]. This implies the following. PROPOSITION

2.14. There is a space X such that ed(X) < ed(X)/or every compactifi-

cation X ofX. Consequently we cannot assume in Corollary 2.1 that the space X^^^ is compact. This still leaves the following question open. PROBLEM 2.2. Does there exist a compact space X^^^ such that Qd{X^^^) = [L] and which contains topological copy of every compact space Y with ed(y) ^ [L]?

For finitely dominated complexes, the situation is simpler. A helpful observation here is the following statement. 2.2 ([55]). Let L be a finitely dominated connected locally compact complex. Then the following conditions are equivalent for any {not necessarily normal^) space X: (i) L G AE(X). (ii) L G AE(ySX), where fiX denotes the Stone-Cech compactification ofX.

THEOREM

Although Theorem 2.1 allows us to establish several interesting facts, the lack of properness of the [L]-soft map constructed in this theorem does not produce extension dimension ^ See Definition 5.1.

316

A. Chigogidze

preserving compactifications or compact universal spaces of a given extension dimension. Our next statement provides SL proper [L]-invertible (as it follows from Proposition 2.16 there are no proper [L]-soft maps of [L]-dimensional spaces) map. 2.3. Let L be a finitely dominated complex and X be a (Polish) space. Then there exists a proper [L]-invertible map fx '-Yx -^ X, where Yx is a (Polish) space such that ed(yx) = [L]. If, in addition, A is a closed subset of X such that ed(A) < [L], then we may assume that fx/fx (^) • fx ^^^ -> A is a homeomorphism. THEOREM

PROOF. Let A denote the set of all maps {ft: t eT} such that domain Dom(/^) is a Polish space, ed(Dom(/^)) ^ [L] and Ran(/r) c /^. Let Y = 0{Dom(/,): t e T]. Clearly, ed(y) ^ [L]. Consider the map f :Y -^ /^, which coincides with the map f on Dom(/^) for each t eT, and its extension fif \pY -> I^. Since L is finitely dominated, by Theorem 2.2, we have td(PY) = ed(Y) < [L]. Consequently, by [53, Theorem 4.4], pY is the limit space of a Polish spectrum Sy = {Ya^qa, A] consisting of compact metrizable spaces Ya such that ed(Ya) ^ [L] for each a e A. Since I^ is a metrizable compactum there exist [51] an index a e A and a map fa'-Ya -^ I^ such that Pf = /^^^Q,. We claim that the map fa -> / ^ is [L]-invertible. Indeed, let ^: Z -^ / ^ be a map such that ed(Z) ^ [L]. By the definition of A, there is an index t e T such that ft = g. Let it: Dom(/^) = Z ^^ Y denote the corresponding embedding. Clearly f = fit. Consequently, the composition h =qait''Z ^^ Ya lifts the map g, i.e., fc^h = g. This shows that fa is [L]-invertible. If X is an arbitrary (Polish) space, then embed X into I^ and consider the [L]-invertible map fa'-Ya -> / ^ constructed above. Clearly, ed(FA:) ^ ^d(Ya) ^ [L], where Yx = f~^(X). It only remains to note that the map fx = fa/Yx'-Yx ^^ X is proper and L-invertible and ed(rx) = [^]. • PROPOSITION 2.15. LetL be a finitely dominated complex, ed(X) ^ [L] and f:X-^Y be a map of a Polish space X into a metrizable compactum Y. Then there is a map f :X ^^ Y such that X is a metrizable compactification of X with ed(X) ^ [L] and

f = f/x. COROLLARY 2.3. Let L be a finitely dominated complex. Then there exists a metrizable compactum X^^^ such that Qd(X[i]) = [L] and the following conditions are equivalent for any space X: (i) ed(X) < [LI (ii) X admits an embedding into X^^\ PROOF. Let / : X^^^ -> / ^ be an [L]-invertible map constructed in Theorem 2.3. We claim that X^^^ is the desired compactum. First we show that Qd(X^^^) = [L]. Assuming that this is not true, and having in mind that Qd(X^^^) ^ [L], we conclude that there is a complex K such that td(X^^^) < [K] < [L]. This, by the definition of the order on extension types, means that there is a Polish space Y such that L € AE(y) but K ^ AE(y). Embed Y into / ^ . Since ed(F) ^ [L] and since / is [L]-invertible, there is a map g:Y -^ X^^^ such that

Infinite dimensional topology and shape theory fg = idy. Obviously, g is an embedding and hence ed(y) = ed(^(F)) < ed(Xt^^) ^ [K], which is impossible by the choice ofY{K^ AE(7)). Thus, ed(X[^]) = [L]. (i) => (ii). We may assume that X C I^. Since ed(X) ^ [L] and since / is L-invertible, there is a map g:X -^ X^ such that fg = id^. Clearly, g is an embedding. The implication (ii) =^ (i) is trivial. D The following corollary does not contradict Proposition 2.14 and follows from Theorem 2.2. COROLLARY 2.4 (LA. Shvedov, see [82, 111]). Let L be a finitely dominated complex and ed(X) ^ [L]. Then there exists a compactification XofX such that ed(X) ^ [L]. PROPOSITION 2.16. There is no [L]-softmap f :X ^^ Y where X is a compactum with ed(X) ^ [L] and Y is a connected ANE-compactum with Qd(Y) > [L]. PROOF. First consider the case of nonconnected L. By Example 2.1(vii), [L] = [S^]. Then dimX = 0 and, by [53, Remark 5.10] and [51, Corollary 6.1.27], / is open. Therefore Y must also be zero-dimensional and this contradicts our assumptions. Suppose now that L is a connected complex. By Example 2. l(viii) and Proposition 2.13, the fibers of / are connected. This in turn implies that X is connected. By the following proposition, / is a cell-like map. By [80], / is approximately invertible and hence ed(y) < ed(X) < [L]. Contradiction. D

PROPOSITION 2.17 ([61]). Every non-constant [L]-soft map f :X -> Y, where X is a connected compactum such that ed(X) ^ [L], is cell-like.

2.4. Z-sets 2.4.1. Limitation topology on spaces of maps. The collection of all open (countable) covers of a space X is denoted by cov(X). Two maps / , ^ : 7 ^- X are called U-close (notation: (/, g) < U), where U e cov(X), if for each point y eY there exists an element U eU such that /(y),g(y) G U. Denote by B(fU) the collection of all ZY-close to / maps. The limitation topology on the set C(Y, X) of all continuous maps of Y into X is now defined as follows. A set G c C(Y, X) is open if for each f e G there is an open cover U e cov(A') such that B(f, U) c G. Note that the sets B{f, hi) are not necessarily open. We say that a map / : 7 ^- X is a U-map, where U e cov(7), if there exists an open cover V e cov(X) such that the cover f~\V) = {/"^ (V): V e V] refines U. LEMMA

2.1.

Let U e cov(y). Then the set of all U-maps is open in C{Y, X).

LEMMA 2.2. Let p be a complete metric on a Polish space Y and suppose a sequence {Un'. n e N} c. cov(7) is chosen so that diam^ U < ^ for each U eUn and each n e N. Then a map f :Y ^^ X is a closed embedding (i.e., an embedding with closed image) if

317

318

A. Chigogidze

and only if f is a Un-map for each n e N. In particular, the set of all closed embeddings ofY into X is a G8-subset ofC(Y, X). Every bounded metric p on a space X generates the sup-metric jo on C(Y, X): p(f,g)=sup{p{fiy),g(y)):

yeV},

f,geC{Y,X).

The topology generated by the metric p on the set C{Y, X) is called the topology of p-uniform convergence. Let Metr(Z) denote the collection of all bounded metrics on a space X compatible with the topology of X. The collection [p: p G Metr(Z)} generates the topology of uniform convergence with respect to all bounded metrics on C(F, X). Obviously, this topology is stronger than the topology of p-uniform convergence for each p G Metr(Z). LEMMA 2.3. The limitation topology coincides with the topology of uniform convergence with respect to all bounded metrics.

The following statement provides another description of the Hmitation topology. LEMMA 2.4. Let p G Metr(X) and f G C ( 7 , X). The collection {Bp(f,a): a e C(X, (0, oo))}, where B,(f,a) = {g e C(Y,Xy. p(f(y),g(y)) ^ aif(y)) for each y eY}, forms a local basis at f in C(Y, X). The following statement expresses a very important and frequently used property of the limitation topology. PROPOSITION 2.18. Let X be a Polish space, F be a subspace of the space C(Y, X) and the set Gn be open in C{Y,X) for each n e N. If the intersection Gn (^ F is dense in F for each n e N, then F c clc{Y,x)i(f^ Gn) H Fp), where Fp denotes the closure of F in the topology of p-uniform convergence and p is any bounded metric on X. In particular, C(Y,X) has the Baire property. Proposition 2.18 can be used to characterize maps between Polish spaces which are approximable by homeomorphisms. Such maps are called near-homeomorphisms. More formally, a map / : 7 ^^ X is a near-homeomorphism if, given U G cov(X), there is a homeomorphism hn'.Y -> X which is Z//-close to / . THEOREM 2.4 (Bing's Shrinking criterion). A map f :Y -^ X between Polish spaces is a near-homeomorphism if and only if f(Y) is dense in X and the following condition is satisfied:

(*) For each U G cov(F) and V G cov(X) there exist an open cover W G cov(X) and a homeomorphism h:Y -> Y such that fh G B(f, V) and hf~^(yV) < U. COROLLARY 2.5. A closed surjection f'.Y-^X between Polish spaces is a nearhomeomorphism if and only if the following condition is satisfied:

Infinite dimensional topology and shape theory

319

(**) For each hi G cow{Y) and each V G cov(X) there exists a homeomorphism h:Y -^ Y such that fh G B(f, V) and the collection {hf~^(x): x e X} refines U. 2.4.2. Definition of {strong) Z-sets. As usual a closed subset A of a space X is said to be a Z-set in X if the set {/ G C ( X , X): f(X) n A = 0} is dense in the space C(X, X). If the set {/ G C(X, X): clx / ( X ) H A = 0} is dense in C(X, Z), then we say that A is a strong Z-set in X. The concepts of the Z-set and the strong Z-set differ even for very simple spaces [22]. Consider the subset ([0,l]x{0})u(u{{i)x[0.1])) of the plane:

(0,0)

(^0)

a,o)

(1,0)

It is not hard to see that X is a AE-space. Moreover, X is even locally compact at each point other than the origin (0,0). For each ? € [0,1) consider a map ht.X—>X determined by the following properties: - h,{l,0) = (l,0). -/^.(^,l) = (i,l). - ht linearly shrinks the segment [0,1] x {0} onto the segment [t, 1] x {0}. - ht linearly expands the segment {^} x [/, 1] onto the segment {^} x [0,1]. - ht linearly maps the segment {^} x [0, ^] onto the segment [^, ^ + —^] x {0} (sending ( i O o n t o ( i 0) and (^,0) onto (/ + ^ , 0 ) ) . Clearly, the identity map idx is in the closure (in the space C(X, X)) of the set {ht: t e (0,1)}, and since ht(X) c X - {(0,0)} it follows that the set {(0,0)} is a Z-set in X. At the same time {(0,0)} is not a strong Z-set in X. To see this, observe that X — G is disconnected for sufficiently small neighborhoods G of the point (0,0). LEMMA 2.5. Let L be a finitely dominated complex and X be an PiNE{[L\)-space such thated(A) < [L]. For each open cover U G COV(X) there exist a locally compact space Ku with td{K]j) ^ [L] and two maps a:X -^ Ky and P : Ku -^ X such that the composition fia is U-close to idx-

320

A. Chigogidze

PROOF. Assume that X is closed in R^ and consider a proper [L]-invertible map p.Y ^ R"^ where 7 is a PoHsh space with ed(y) ^ [L]. Also let q:p~^{V) -^ X be an extension of the map p/p~^ {X) and W = {U^: U eU}bQa. collection of open subsets of V such that

U'nX = U and p~\u')^q~\U)

for each (/G ZY.

Since S = \^{U^: U eU] isa. Polish ANE-space there exist a locally compact polyhedron K and two maps y:S-^K and 8:K -^ S such that 8y is ZY'-close to id^. By Theorem 2.3, there is a proper [L]-invertible map h : K^ -^Kofa. (locally compact) space KK such that cd(Ki() < [L]. The [L]-invertibility of h coupled with the fact that e(X) ^ [L] guarantees that there is a map a:X ^^ Ku such that ha = y/X. The [L]-invertibility of p coupled with the fact that e{Kii) < [L] implies the existence of a map fi': Kjj -^ p~\L) such that PP^ = 8. Let P = qfi'. It only remains to observe that afi is indeed U-c\ost to idx. • PROPOSITION 2.19. Let L be a finitely dominated complex, X^^^ be a universal compactum in the class of spaces whose extension dimension does not exceed [L] and {X[ : k ^ 0)} be a countable closed base of X^^^ containing intersections of its finite subcollections. Then the following conditions are equivalent for a closed subset A of a Polish ANE([L])-space X: (i) The set {/ G C(x[^\ X): f(x[^^) r]A = 0] is dense in the space C(X[^\ X) for each k eco. (ii) The set {/ G C(Y, X): f(Y) H A = 0} is dense in the space C(Y, X) for each compact space Y with ed(y) ^ [L]. (iii) The set {/ G C{Y, X)\ f(Y) fl A = 0} is dense in the space C{Y, X) for each locally compact space Y with ed(7) < [L]. (iv) A is a Z-set in X. (i) ^ (ii). Let / : F ^- X be a map defined on a compactum Y such that ed(y) ^ [L]. We may assume that Y c X^^^. Since ed(X[^^) = [L] and since X is an ANE([L])space, there is a A: G ct> such that F c x [ and / admits an extension f '.x\. ^ ^- X. By (i), / can be approximated by maps g: x[^^ -^ X such that g(X^) fl A = 0. Then the map g = g / Y \Y ^^ X is an approximation of / and g{Y) Pi A = 0. (ii) =^ (iii). Suppose now that 7 is a locally compact space such that ed(F) < [L]. Since Y is a-compact, we can write Y = {J[Yk\ k = \,2,...} where each Yk is compact. Let us show that for each A: = 1, 2 , . . . the set PROOF,

Bk = [feC{Y,xy

/(y^)nA = 0}

is dense in the space C{Y, X). Consider a map f '.Y ^^ X and an open cover U G cov(X). Consider an open cover V G cov(Z) satisfying the condition of (*)[L] of Proposition^ 2.27 relative to U. By (ii), there is a map g'\Yk -^ X such that g' is V-close to f/Yk and There is no logical inconsistency here. Proof of Proposition 2.27 does not depend on Proposition 2.19.

Infinite dimensional topology and shape theory

321

g\Yk) f^A = &. Consequently, by the choice of V, the map g^ has an extension g:Y -> X which is ^/-close to / . Clearly, g{Yk) Pi A = 0. Next observe that the set Bk is open in C{Y, X) and, consequently, by the Baire property of the space C(Y, X) (see Proposition 2.18), the intersection {feCiY^Xy.

f(Y)nA

= 0} =

f]{Bk:k=h2,...}

is also dense in C(Y, X). (iii) =^ (iv). Consider a map f :Y ^^ X and an open covert/ e cov(X). Let V € cov(X) be a star-refinement ofU. For an open cover f~^(V) e cov(7), by Lemma 2.5, there exist a locally compact space K and maps a:Y -^ K and fiiK ^^ Y such that the composition fia is /~nV)-close to idy. By (iii), there is a map g^: K ^^ X which is V-close to the composition ffi and such that g^(K) fl A = 0. Let g = g'a. It is easy to check that g and / are Z//-close and g{X) fl A = 0. The implications (iv) =^ (i) is trivial. D PROPOSITION 2.20. Let L be a finitely dominated complex, X^^^ be a universal compactum in the class of spaces whose extension dimension does not exceed [L] and {X[ : k e 0)} be a countable closed base of X^^^ containing intersections of its finite subcollections. Then the following conditions are equivalent for a closed subset A of a Polish A^^{[L])-space X\ (i) The set

[fec{^{xf'•.keco],x)•.o\xf[^[xf'•.keco])r^A^^ is dense in the space C ( 0 { Z [ : A: G a;}, X) (ii) The set [f ^ ^ ( 0 { ^ ^ - ^ ^ ^ } ' ^ ) - ^^x f(^{Yk:

/: G o)}) n A = 0

is dense in the space C{@[Yk'. k G CL>},X) for any compact spaces Yk with ed(Yk) ^ [L], keco. (iii) The set { / G C ( F , X ) : c l x / ( F ) H A = 0}

is dense in the space C(Y, X) for each locally compact space Y with ed(7) ^ [L]. (iv) A is a strong Z-set in X. PROOF. In order to prove the only nontrivial implication (ii) ^ (iii), represent Y as the countable union Y = [J{Yk: k e oj] of compact subsets such that Yk c int(yjt+i) for each keco. Let Ti = ]J{Y2k-ini(Y2k-i): k£oj}UYo and T2 = \J{Y2k-^i -int(720: k G CO}. By (ii) and Proposition^ 2.27, the sets A/ = {f e C(Y, X): clx f(Ti) n A = 0}, / = 1,2, are See footnote 5.

322

A. Chigogidze

both dense and open in C(Y, X). Finally, by Proposition 2.18, their intersection Ai fl A2 is also dense in C(y, Z). D COROLLARY

2.6. Every Z-set in a locally compact ANE([L])-5/?ac^ is a strong Z-set

in it.

2.5. Strongly [L]-universal spaces In this subsection we discuss the main objects of "infinite dimensional topology modulo L". There are two types of such spaces: /x^^^ - Menger compactum modulo L, and v^^^ - NobeHng space modulo L. 2.5.1. Locally compact case. Below we construct Menger compacta /x^^^ modulo L. DEFINITION 2.4. For a given complex L we say that a space X satisfies the disjoint disks property modulo L (notation: [L]-DDP) if the set

{/ G C(Y X {0,1}, Xy. f(Y X {0}) n f(Y X {1}) = 0}, where 7 is a compactum such that ed(F) < [L], is dense in the space C(y x {0,1}, X). DEFINITION 2.5. For a given complex [L] we say that a locally compact space X is strongly [L]-universal for compact spaces if for any locally compact space B, its closed subspace A, any open cover Z// e cov(X) and for any proper map f: B ^ X, the restriction f/A of which is a Z-embedding, there is a Z-embedding g.B^-X which is Z//-close to / and such that g/A = f/A. PROPOSITION 2.21. Let L be a finitely dominated complex and X be a {locally) compact ANE([L])-5'/7^ce. Then the following conditions are equivalent. (i) X has the [L]-DDP. (ii) X is strongly [L]-universal for compact spaces. COROLLARY

2.7. Every point in a locally compact ANE([L])-space with [L]-DDP is a

Z-set. Later we will see that the proof of Theorem 2.6, which through the inverse limit construction produces nonlocally compact spaces v^^^ discussed above, cannot, without changes, be adapted to a locally compact situation. Reason - the nonexistence of proper [L]-soft maps raising extension dimension (Proposition 2.16). On the other hand, the [L]-invertibility of (proper) projections (such maps do exist - Theorem 2.3) of an inverse sequence consisting of ANE([L])-spaces does not guarantee that the limit is also an ANE([L])-space. The class of maps which is sufficient for our purposes is introduced in the following definition.

Infinite dimensional topology and shape theory

323

2.6. A map f :X ^^ Y is said to be approximately [L]-soft, if for each space B with ed(B) < [L], for each closed subset A of it, for an open cover U e cov(F), and for any two maps g'.A^^X and h.B^^Y such that fg = h/A, there is a map k: B ^^ X satisfying the conditions k/A = g and the composition fk is Z//-close to h. DEFINITION

REMARK 2.2. Basic examples of approximately [L]-soft maps are maps between ANE([L])-compacta whose nontrivial fibers are AE([L])-compacta.

The standard argument [78] proves the next statement. 2.22. Let S = {Xn, Pn'^^ ,co} be an inverse spectrum consisting of (locally) compact ANE([L])-spaces and approximately [L]-soft (proper) projections. Then the limit lim<S is an ANE([LY)-space and each limit projection pn :lim5 -^ Xn is approximately [Lysoft.

PROPOSITION

The following construction [78] is based on an idea of Dranishnikov [73]. Let Cone(L) be the cone over difinitecomplex L and let A: L x [0,1] ^^ Cone(L) be the quotient map. Consider the subspace T = {((l,t),l): (l,t) e L x [0,1)} of Cone(L) x L and an open neighborhood U of the vertex of the cone in Cone(L). Clearly L = (cl t/ x L) U T is a closed subspace of the product Cone(L) x L. Finally, let ^I'.L ^^ Cone(L) denote the restriction of the projection Cone(L) x L ^ - Cone(L) onto L. The following statement expresses an essential property of the map §L (A.N. Dranishnikov). LEMMA 2.6. The map ^L: L -^ Cone(L) is [L]-invertible and approximately [L]-soft. PROOF. The fact that ^L is approximately [L]-soft follows from Remarks 2.1 and 2.2. Let us show that ^L is [L]-invertible. Let (p:B -^ Cone(L) be a map defined on a compactum B with ed(B) < [L]. Let i/r: Cone(^) — t/ -> ^ be the unique section of the restriction ^L/§z^kCone(L) - U): ?^^ (Cone(L) - U)-> Cone(L) - U. Then the composition

T/r' = ir(p/{(p-^ (Cone(L) - U)): (p'^ (Cone(L) - U) ^ K is a lifting of (p/((p-\Cone(L) - U)). Next consider the set (p~\clU) c 5 and A = (p~\clU - U) ^ (p~\c\U). ed(<^~^ (cl U)) ^ [L], the composition (p-^(clU-U)^

Since

Cone(L) x L ^ L

has an extension ^\Q\U -^ K. Then the map cp.B ^^ L defined by letting -(b) = I '^'^^^' '^^ ^ (p-\Com(L) - U) ^ \((p(b),ir(b)), ifbecW, is a lifting of (^.

D

324

A. Chigogidze

LEMMA 2.7. Let L be a finite complex^ X a (locally) finite polyhedron and f : A ^ L a simplicial map defined on a subpolyhedron AofX. Then there is an [L]-invertible approximately [L]-soft (proper) simplicial map p = P(x,Aj) • Y{X,AJ) -^ X satisfying the following conditions'. (i) ^(X,A,/) is a (locally) finite polyhedron. (ii) The composition fp/p~^ (A): p~^(A) -^ L admits an extension over Y(X,AJ)' PROOF.

Extend / to a simplicial map / : X ^- Cone(L) and consider the diagram HX,A,f)-

^L HL

f where

F(X,A,/)

-^ Cone(L)

is apuUback, i.e.,

Y^x,A,f) = {(x,z)eXxL:

f(x) = ^L(z)}

and the maps p : Y(x,A,f) -^ ^ and q : Y(X,AJ) ^^ projections XxL^X

and

L

are restrictions onto Y(x,A,f) of the

XxL^L,

respectively. It is immediate that the map p is also [L]-invertible and approximately [L]-soft (it has same fibers as ^i). Next observe that fp/p~^(A) = ^iq/p~^(A). Consequently the composition Y{x,Aj) —> L ^^ Cone(L) x L is the desired extension of

fp/p~^(A).

Pr2

D

Proposition 2.22, Lemma 2.7 and considerations of [49,79] imply the following important result. 2.23. Let L be a finite complex and X be afinitepoly hedron or a compact Hilbert cube manifold. Then there exists an [L]-invertible and approximately [L]-soft map fx ' l^x ^ satisfying the following conditions: PROPOSITION

(i)

/X^^^GANE([L]).

(ii) ed(/x^^b = [LI (iii) For any map f \B ^^ p}^ , where B is a compact space with td(B) ^ [L], and for any open cover U G cov(/x^ ) there is an embedding g'.B^^v^^ which is U-close to f and such that gf]^ = ffj^ .

Infinite dimensional topology and shape theory

325

In particular, letting X = {pt}, we obtain the following statement. 2.5. There exists a compact space p\^^ satisfying the following conditions: (i) p\^^ is a compact AE([L])-space. (ii) ed(/x[^l) = [LI (iii) /x^^^ is strongly [L\-universal for compact spaces.

THEOREM

2.5.2. Nonlocally compact case. We start with the following two definitions. DEFINITION 2.7. For a given complex L we say that a space X satisfies the strong discrete approximation property modulo L (notation: [L]-SDAP) if the set

| / e C ( ^ 0 { y ^ : ke(o},x\.

{f(Yk): J^ G (w} is discrete in x | ,

where Yk, k e co, is a compactum such that td(Yk) ^ [L], is dense in the space

CmYk:keco},X). DEFINITION 2.8. For a given complex L we say that a Polish space X is strongly [L]-universalfor Polish spaces if for any Polish space B, its closed subspace A, any open cover hi e cov(X) and any map f: B -> X such that the restriction f/A is a Z-embedding, there is a Z-embedding g:B -^ X which is Z//-close to / and such that g/A = f/A. COROLLARY 2.8. Let L be a finitely dominated complex and X be a Polish ANE([L])space satisfying [L]-SDAP. Then the following conditions are equivalentfor a closed subset A of X: (i) A is a Z-set. (ii) A is a strong Z-set. COROLLARY 2.9. Let L be a finitely dominated complex. Every compact subset of a Polish ANE([L])-space satisfying [L]-SDAP is a strong Z-set. PROPOSITION 2.24. Let L be a finitely dominated complex and Xbea Polish ANE([L])space. Then the following conditions are equivalent. (i) X has the [L]-SDAP. (ii) X is strongly [L]-universal for Polish spaces. PROOF. We prove only the absolute case. The relative version requires additional technical considerations. Step L First we show that for each locally compact space Y with ed(y) < [L] the set of closed embeddings of Y into X is a dense (and, according to Lemma 2.2, G^-) subset of the space C{Y, X). Write Y = Y\ U 72, where Y\ and F2 are unions of discrete (in Y) collections of compacta. By (/) and Proposition 2.27, the set

d = {/ € C(y, Xy f(Yi) is closed in X] is dense and open in C{Y, X), / = 1, 2. By Proposition 2.18, the set D = Ci fl C2 is still open and dense and obviously consists of maps with closed in X images.

326

A. Chigogidze

Next consider a countable open basis {Uk'. k eco} of Y. Applying the above argument to each of the sets Y — Uk,^^ conclude that the set Dk = {f e C{Y, X)\ f(Y - Uk) is closed in X] is open and dense in C{Y,X),ke co. Since every open subset of Y is a-compact, Corollary 2.9 and Proposition 2.18 imply that Tm,n = [fe

C ( 7 , Xy.

f{Um)

H f{Un)

= 0}

is a dense and G^-subset of C(7, X) for each m,n e o) with Um (^ Un = 0. Applying Proposition 2.18 once again we see that the intersection

Dnf]{Dk:

keco}nf]{Tm,n:

m,nea)}

is a dense and G<5-subset of C(Y, X), It only remains to note that this intersection consists of closed embeddings of Y into X. Step 2. Now consider the general situation. Let / : 7 ^- X be a map of a Polish space Y such that ed(F) ^ [L] and let U G cov(y). By Corollary 2.1, we may assume that 7 is a closed subspace of a Polish AE([L])-space X^^\ Since X is an ANE([L])-space, there is an extension / : G ^- X of the map / onto an open neighborhood G of 7 in X^^\ Let U denote an open cover of G such that St(t{)/X refines U. By Proposition 2.11, G is a Polish ANE([L])-space and, hence, by Lemma 2.5, there exist a locally compact space K^ with ed(^^) ^ [L] and two maps a : G -> K^ and P: K^ -^ G such that a^ is U-c\osc to ida- Next consider a countable collection of compact subsets {Aki k e co} of X with the following property: any closed subset A of X such that A c X — |J{A^: k eco} is 3. Z-set in X (the existence of such a collection follows from Proposition 2.19(i)). According to step 1 and Corollary 2.9, the map fP : K^ -^ X can be approximated by a closed embedding g: K^ -^ X such that g(K^) H |J{A^: k eco = 0}. It is easy to see that the map g = ga:Y -^ X approximates / , is a U-map and /(Y) c U{A^: k e co). This shows that the set of U-msips of Y into X with images in X — [J{Ak: k e co} is dense and (by Lemma 2.1) a G^-subset of C(Y, X). Finally, Lemma 2.2 guarantees that the set of closed embeddings of Y into X with images in X — lJ{Ajt: k e co} is dense and a G^-subset in C(Y, X). By the choice of {A^: k e co}, every such embedding is a Z-embedding. D THEOREM 2.6. Let L be a finitely dominated complex and X be a Polish A(N)E([L])space. Then there exists an [L]-soft map / x • ^x ^ ^ satisfying the following conditions: (i) y[,^UA(N)E([L]).

(ii) ed(v^'^b = [LI (iii) For any map f \B ^^ v\^\ where B is a Polish space with td{B) < [L], and for any open cover U G cov(y^ ), there is a closed embedding g'.B-^v\^ which is hi-close to f and such that gfj^ = ffj^ .

Infinite dimensional topology and shape theory

327

Let XQ = X. Let Xi be a Polish space which admits an [L]-soft map h\:Xi -> XQ X R^ onto the product XQ X R ^ and ed(Xi) ^ [L] (the existence of Zi and h\ is guaranteed by Theorem 2.1). Let p^ = noh \, where jto'.Xox R^ -^ XQ denotes the natural projection onto the first coordinate. In general, if a Polish space Xk with ed(X^) ^ [L] has already been constructed, as the space Xk-\-\ we take any Polish space which admits an [L]-soft map hk-\-i: Xk-\-i -^ XkX R^ and such that Qd(Xk) ^ [L]. We also define the map PI'^^ as the composition p^'^^ = 7Tkhk-\-\, where 7Tk:XkX R^ ^^ Xk denotes the projection onto the first coordinate. Therefore we have an inverse sequence S = {Xk, p^'^^, co}. Let y^^ = lim5 and f^^^ = po- By Proposition 2.6, ed(v^^) ^ [L]. Considerations similar to the proof of [51, Proposition 5.1.11] guarantee that the limit projection/j. :Vx ~^ ^ has the desired properties. Observe also that /j^ (being the limit projection of the inverse sequence with [L]-soft short projections) is [L]-soft. In particular, by Proposition 2.13, y^^^ G A(N)E([L]) and ed(y^^b = [L]. D PROOF.

Letting X = {pt} in the above theorem we obtain the following important statement which significantly strengthens Corollary 2.1. THEOREM 2.7. Let L be a finitely dominated complex. Then there exists a space v^^^ satisfying the following conditions: (i) yf^^ is a Polish A^{{LY)-space. (ii) ed(y[^J) = [L]. (iii) yf^^ is strongly [Ly universal for Polish spaces.

Different constructions (including direct geometrical ones) may produce at first glance different spaces satisfying the conditions of Theorem 2.7. For instance, it is not clear whether the spaces yL and yL \. are homeomorphic.

2.6. [L]-homotopy We need to develop an adequate homotopy language for ANE([L])-spaces. The key here is the following definition. DEFINITION

2.9.

Two (proper) maps /o, / i :X -> Y are said to be (properly) [L][L]

[L]

homotopic (notation: /o — / i , /o — ^ / i ) if for any (proper) map h:Z -^ Z x [0, 1], where Z is a space with ed(Z) ^ [L], the composition (/o 0 f\)h/h~\X x {0, l}):h~\X x {0,1}) -> y admits an (proper)extension H:Z ^^ Y. 2.7. Observe that: (i) [L]-homotopic maps are [^]-homotopic for each K with [K] < [L]. (ii) If X is an AE([L])-space, then idx is [L]-homotopic to a constant map. (iii) The concept of [5"]-homotopic maps between complexes coincides with the notion of n-homotopic maps introduced by J.H.C. Whitehead in [151]. It should be emphasized that later, certain reasons within algebraic [15], [16, p. 124], [125]

EXAMPLE

328

A. Chigogidze

and geometric [20,43,44,46,51] topology led to a shift of scale and Whitehead's (n + 1)-types are now commonly referred as «-types. It seems now more appropriate to change the terminology back. Thus everywhere below, the Az-homotopy denotes the 5"-homotopy, i.e.. Whitehead's n-homotopy types. 2.25. Let L and K be connected finite complexes and {K^ < [L]. Then the identity map id A: is not [L]-homotopic to a constant map.

PROPOSITION

PROOF. Consider an [L]-invertible and approximately [L]-soft map f :X ^^ Conc(K) where X is an AE([L])-compactum with cd(X) = [L] (Proposition 2.23). Now, assuming that id/: is [L]-homotopic to a constant map, we see that there is a map g:X -^ K such that g/f~\K) = flf-\K) (here K is identified with the bottom, K x {0}, of the cone Cone(^)). Take a compactum Y such that ed(7) = [L]. Since [K^ < [L], there exists a map Of: Yo -> K, defined on a closed subspace YoofY, which does not admit an extension over y. Since ed(yo) ^ [L] and / is [L]-invertible, there is a map P'.YQ -^ X such that a = fp. Since X is an AE([L])-compactum and ed(F) < [L], there is a map y :Y ^^ X such that p = y/Yo. Then the composition gy :Y ^^ K produces an extension of a. Contradiction. Thus id/^ is not [L]-homotopic to a constant map. D

According to Proposition 2.7, the extension dimension of complexes coincides with the usual dimension. Nevertheless, as follows from the above discussion, the concept of [L]-homotopy differs from the classical notions of homotopy or n-homotopy, even for maps between complexes. Indeed, let /i ^ 1 and L be a connected complex such that [5"] < [L] < [5"+M (Example 2.4(iii)). By Remark 2.1, Example 2.7(ii) and Proposition 2.25, we see that [L]

[5"+l]

idL — const, but id/, ^

const.

Similarly, idsn ^ const, but id^Ai g^ const. Later we make use of some natural concepts associated with [L]-homotopy. Here are their definitions. DEFINITION 2.10. A (proper) map / : X ^^ 7 is a (proper) [L]-homotopy equivalence if there is a (proper) map g:Y ^^ X such that the compositions gf and fg are (properly) [L]-homotopic to idx and idy, respectively. DEFINITION 2.11. Let ZY G cov(y). We say that maps / , g: X -> 7 are U-{Lyhomotopic if for any map /z: Z ^- X x [0,1], where Z is a space with ed(Z) ^ [L], the composition (/o e f\)h/h-\X X {0, \}):h-\X x {0,1}) -^ Y admits an extension H\Z^Y such that the collection [H{h-\{x] x [0,1])): xeX} refines Z^.

Infinite dimensional topology and shape theory

329

DEHNITION 2.12. A map / : X ^- F is difine [L]-homotopy equivalence if for any U e cov(y) there is map gu'.Y ^^ X such that the composition fgjj is Z//-[L]-homotopic to idy and the composition gjjf is /~H^)-[^]-Nomotopic to idx-

2.6.1. [L]-homotopy properties of ANE([L])-spaces PROPOSITION 2.26. Let L be a finitely dominated complex and U e cov(Z) be an open cover of a Polish ANE([L])-space X. Then there exists an open cover V G cov(X) such that any two V-close maps of any space into X are U-[L]-homotopic. PROOF. We may assume that X is a closed subspace of R^. Fix a proper [L]-invertible map f'.Y^R"^ such that ed(y) ^ [L] (see Theorem 2.3). Since X e ANE([L]), the restriction f/f~^{X): f~^{X) -^ X admits an extension g:U ^^ X over a neighborhood U of f~^(X) in Y. The properness of / allows us to assume that U = f~^(V) for some neighborhood V of X in R"^. Consider an open cover Z^/' = {V - f(g~^ (X - U)): U eU] of V. Since V is a Polish ANE-space, there exists an open cover V' e cov(y) such that any two V^-close maps into V are ZY^-homotopic. Let^ V = V^X and let us show that V is the desired open cover of X. Consider a space Z and any two V-close maps Qfo, oii'.Z ^^ X. Then, by the construction, there is a homotopy / / : Z x [0,1] ^- V such that H(z, i) = ai(z) for each (z, /) € Z x {0,1} and the collection {//({z} x [0,1]): z e Z} refines U\ Now consider a map ^ : Z -^ Z x [0, 1] where Z is a space such that ed(Z) ^ [L]. Since ed(Z) ^ [L], the [L]-invertibiHty of / guarantees the existence of a map G:Z ^^ U such that fG = Hfi. Finally, \ti F = gG.Z ^^ X. One can easily verify that F is indeed an extension of (ofo 0 a\)P/^~^{Z x {0,1}) and that the collection {F{fi-\{z} X [0,1])): zeZ) refines Z^. Thusofo and a\ are Z//-[L]-homotopic. D

The following two statements (their proofs follow the proof of Proposition 2.26) are versions of the "[L]-Homotopy Extension Theorem". PROPOSITION 2.27. LetL be a finitely dominated complex and X be a Polish ANE{[L])space. Then for each U G cov(X), there exists V e cov(X) refining hi, such that the following condition holds:

(*)L For any space B with cd(B) ^ [L], any closed subspace A ofB, and any two V-close maps f,g:A-^X such that f has an extension F: B ^^ X, it follows that g also has an extension G: B -^ X which is U-[L]-homotopic to F. PROPOSITION 2.28. Let L be a finitely dominated complex and X be a Polish ANE([L])space. Suppose that A is closed in a space B with cd(B) ^ [L]. If maps f g: A ^^ X are [L]-homotopic and f admits an extension F: B ^ X, then g also admits an extension

G: B ^ X, and it may be assumed that F — G. 2.13. We say that a compactum X is a UV^^^-compactum if the following condition is satisfied: DEFINITION

V/X = {V'nX:

V'eV}.

330

A. Chigogidze

- For any embedding of X into an ANE([L])-space Y with ed(F — X) ^ [L], for any neighborhood U of X inY there is a smaller neighborhood V such that the inclusion V "-> U is [L]-homotopic (in U) to a constant map. Not surprisingly, we implicitly have already met f/V^^^-compacta. Indeed, the fibers of proper approximately [L]-soft maps (between ANE([L])-spaces) are -compacta. This suggests another, perhaps more familiar, name for approximately [L]-soft maps -maps. The standard argument [102] proves the following statement. 2.29. Let X ^ Y be a map between ANE{[L\)-compacta and td{Y) < [L]. Then the following conditions are equivalent. (i) / is a UV^^^ -map. (ii) / is approximately [L]-soft. (iii) / is a fine [L]-homotopy equivalence.

PROPOSITION

The following result provides an internal characterization of ANE([L])-spaces. Its proof is standard (see [24] for details). THEOREM 2.8. Let [L] ^ [5"]. Then the following conditions are equivalent for any space X: (i) X is an ANE([L])-space. (ii) X is locally [L]-contractible, i.e., for each neighborhood Ux of any point x G X, there exists a smaller neighborhood Vx such that the inclusion Vx ^^ Ux is [L]-homotopic (in Ux) to a constant map.

How essential is the requirement [L] ^ [5"]? What happens if the extension type [L] is larger than any [S^] (which, according to Example 2.1(iv), means that [L] = [{pt}]) or is not comparable with any [5"] (this possibility, according to Example 2.5, also can be realized)? In both cases the answer is the same - the statement in these cases is not true. Borsuk's example [24] of a locally contractible space which is not an ANE-space serves as a counterexample in both cases. 2.6.2. [L]-homotopy groups. The next step in our program is to define algebraic [L]-homotopy invariants. Here we discuss only [L]-homotopy groups. Everywhere below L stands for a connected finite complex. For each n ^ 0 we consider an ''nth [L}-sphere" 5r"^n, which is nothing else but an [L]dimensional ANE([L])-compactum {jJi\n in the notations of Proposition 2.23) admitting an approximately [L]-soft map onto S^. It can be shown that all possible choices of an [L]-sphere 5r"^j are [L]-homotopy equivalent. This allows us to consider, for each n^ \ and for any space X, the set 7tn (X) = [S?LY ^^t^] ^^ [L]-homotopy classes of maps of the [L]-sphere S!^^ into X. It turns out that this set carries a natural group structure, i.e., for an [L]-homotopy class [ / ] [ L ] ^ [X, F][L], the natural map

Infinite dimensional topology and shape theory

331

is a well defined homomorphism. We call the group jtn (X) the nth [L]-homotopy group of the space X. Even for a complex K, the system {Ttn (K): « = 1, 2,...} of [L]-homotopy groups may differ from the system {7Tn(K): n = l,2,...} of usual homotopy groups. 2.8. (i) If [L] = [{pt}], then the A:th [{pt}]-sphere is (homotopically) the same as the usual sphere 5^, and consequently the system of [{pt}]-homotopy groups of any space X coincides with the system of usual homotopy groups EXAMPLE

7ri(X),7r2(X),.... (ii) If [L] = [5"], then ^th [S'^J-sphere sLni is the same (homotopically) as the usual sphere S^ precisely for A: ^ n — 1. For k^ n, the /:th [^"j-sphere Sl^^n^ is [5"^^]-homotopy equivalent to S^, which in turn (since k^n)is[S^]-homotopy equivalent to a point. Therefore the system of [5""]-homotopy groups of a space X looks as follows 7ri(X),7r2(X),...,7r^_i(X),0,0,...,0,.... What can we say about maps / : Z ^- 7 which induce homomorphisms of all [L]-homotopy groups? Let us consider only spaces with nice local [L]-homotopy structure. For instance, consider ANE([L])-spaces (see Theorem 2.8) of extension dimension not exceeding [L]. In addition, let us assume (in order to avoid technical difficulties which are not essential for the present discussion ... see Problem 2.14) that both X and Y are [L]-homotopy equivalent to spaces which admit (proper) approximately [L]-soft and [L]-invertible maps onto locally finite polyhedra. Let us temporarily call such spaces triangulable ANE([L])-spaces or simply [L]-complexes. THEOREM 2.9. A map between [L]-complexes is an [L]-homotopy equivalence if and only if it induces isomorphisms o/all [L]-homotopy groups.

What does this theorem tell us in the cases [L] = [{pt}] and [L] = [5"]? If [L] = [{pt}], then [L]-complexes are precisely (locally compact) ANE-spaces (this follows from Theorems 3.34 and 3.39 or 3.22 and 3.27). Therefore, according to Example 2.8(i) we obtain Whitehead's well known result. THEOREM 2.10 ([150,151]). A map between ANE-spaces is a homotopy equivalence if and only if it induces isomorphisms of all homotopy groups.

There is another important result of Whitehead which states the following. THEOREM 2.11 ([150,151]). A map between at most n-dimensional ANE(n)-spaces is an n-homotopy equivalence if and only if it induces isomorphisms of the kth homotopy groups for each k ^n — I.

Why are only the first n — l homotopy groups responsible for making a map an n-homotopy equivalence? What is common between Theorems 2.10 and 2.11? How is the last

332

A. Chigogidze

result related to Theorem 2.9? A crucial observation here is that the system of the first n — I homotopy groups is the initial segment of the system of all [5"]-homotopy groups, the remaining part of which always consists of trivial groups (Example 2.8(ii)) and therefore cannot be seen explicitly. Remarking now that [5^^]-complexes are precisely at most n-dimensional LC^"^-spaces (this follows from the results of Section 4), we arrive at the following corollary of Theorem 2.9, which is just a reformulation of Theorem 2.11 but allows us to understand the whole picture. THEOREM 2.12. A map between at most n -dimensional LC^ ~ ^ -spaces is an n -homotopy equivalence if and only if it induces isomorphisms o/all [S^]-homotopy groups.

It is easy to see that the category of complexes and [^S^J-types is isomorphic to the category of sets and their maps. It is known [114,151] that: (a) The category of connected complexes and their [5^]-types can be identified with the category of groups and their homomorphisms. (b) The category of connected complexes and their [5"^]-types is isomorphic to the category of crossed modules. For n > 4 there are partial results characterizing the category of complexes and [5"]-types. See [15] for further information. In the light of these results, the problem of the algebraic characterization of the [L]-homotopy category H O M O T [ L ] becomes very intriguing. 2.3. Characterize algebraically the category of [L]-complexes and their [L]-homotopy types.

PROBLEM

In the case L = {pt} there is an important notion of simple homotopy (see [62] for details). It was shown in [37] that a map f :X -> Y between compact ANE-spaces is a simple homotopy equivalence if and only if there exist a compact ANE-space Z and two cell-like maps a : Z -> X and ^ : Z ^- 7 such that fa^ ^.In the case L = S" there are no simple homotopy obstructions. We conjecture that the same applies to complexes L with [L] ^ [5"]. In the remaining cases, i.e., for complexes the extension types of which are not comparable with any [5"], the situation is not clear. Of course, guided by the above cited result of Chapman, we can formally introduce the concept of simple [L]-homotopy equivalence. DEFINITION 2.14. A (proper) map f:X^Y between ANE([L])-spaces is called a (proper) simple [L]-homotopy equivalence if there exist an ANE([L])-space Z and (proper) f/V^^^-maps a.Z -^ X and ^\Z -^Y such that fa is (properly) [L]-homotopic to^. We specifically ask: 2.4. Let L be afinitecomplex not comparable with any [5"]. Do the classes of [L]-homotopy equivalences and simple [L]-homotopy equivalences differ? More specifically, is it true that there exists an [L]-homotopy equivalence between ANE([L])-compacta of extension dimension not exceeding [L] which is not a simple [L]-homotopy equivalence? PROBLEM

Infinite dimensional topology and shape theory

333

2.7. [Lyshape In this section we define the [L\-shape category (notation: S H A P E [ L ] ) . In the classical case (i.e., L = {pt}) there are at least three alternative approaches to the concept of shape: geometrical, spectral and categorical (see [25,84,115] for complete details). There is also the fourth approach based on Chapman's complement theorem [33]. We will not discuss algebraic aspects of the [L]-shape theory, and instead will concentrate on its geometrical side. Although it is better to have the [L]-shape category defined over the class of all compacta, for technical reasons we prefer to discuss here the category S H A P E [ L ] whose objects are compacta of extension dimension not exceeding [L]. Morphisms a e Mor(X, Y) of this category are defined as maps^ a : IJ{[F, P][L]: P e ANE([L]) and ed(P) < [L]} ^ | J { [ X , P][L]: P e ANE([L]) and ed(F) ^ [L]} satisfying the following two conditions: (i) IfPe ANE([L]) and ed(P) ^ [L], then a([Y, P ] [ L ] ) C [X, P ] [ L ] . (ii) If P, P' G ANE([L]), ed(P) < [ L ] , ed(PO < [^1, V^ e [7, P ] [ L ] , ^ ^ [Y.

P'][L]

and (p G [P, P^][L] are such that (p-i// = V^^ then (pa(\lr) = a(\lr^).

Next we define the [Lyfundamental functor J^m : H O M O T [ L ] -^ S H A P E [ L ] . It acts as follows: T[L](X) = X for any compactum X with ed(X) ^ [L]; if cp e [X, Y][L], then the [L]-shape morphism J^[L]((P) :X->Yis given by the rule: ^[L](V^) = ^(p for each \l/ e [F, P][L], where P is an arbitrary ANE([L])-compactum with ed(F) < [L]. The composition of the functors H[L] (the natural [L]-homotopy functor) and T[L] is denoted by 5[L]: COMP([L]) -^ SHAPE[L] and called the [Lyshape functor (here COMP([L]) denotes the class of compacta whose extension dimension does not exceed [L]). We now arrive at the natural definition. DEFINITION 2.15. Let X and Y be compacta with ed(X), ed(y) < [L]. We say that X and Y have the same [L]-shape (and write Sh[L] {X) = Sh[L] (F)) if X and Y are isomorphic objects of the category S H A P E [ L ] .

Obviously [L]-homotopy equivalent compacta have the same [L]-shape but not vise versa. It is also an easy exercise to show that Sh[L](X) = Sh[L]({pt}) if and only if X is an -compactum. 2.5. A map f :X -^ y is a hereditary [Ly shape equivalence if for each closed subspace A of 7, the [L]-shape morphism generated by the map f/f~\A): f~^(A) -^ A is an [L]-shape equivalence. Do the classes of L^V^^^-maps and hereditary [L]-shape equivalences between compacta of extension dimension ^ [L] coincide? PROBLEM

Is it possible to define a concept of [L]-shape dimension? How about algebraic [L]-shape invariants? Is there a satisfactory approach to the notion of strong [L]-shape? Another interesting question: ^ [Y, P][L] stands for the set of [L]-homotopy classes of maps of Y into P.

334

A. Chigogidze 2.6. Characterize [L]-dimensional compacta with a polyhedral [L]-shape.

PROBLEM

Perhaps the most fundamental geometrical property of [L]-shapes is expressed in the following result (the cases L = {pt} and L = S^ were considered in [33] and in [43,44], respectively). 2.13 (Complement Theorem for /x^^^). Let X and Y be Z-sets in the Menger compactum fi^^^ modulo L. Then the following conditions are equivalent: (i) The complements /x^^^ — X and /x^^^ — Y are homeomorphic. (ii) Sh[L](X) = Sh[i](Y), i.e., X and Y have the same [L]-shape. THEOREM

The proof of this theorem can be carried out under the assumption of the validity of Conjecture 2.8. In fact, the Complement Theorem has a categorical nature. In order to formulate the precise result we first need to introduce the corresponding definitions. Loosely speaking, two proper maps f.g'.X ^^ Y between ANE([L])-spaces of extension dimension < [L] are said to be weakly properly [L]-homotopic if for each compactum B C.Y there is a compactum AC.X such that the restrictions / / ( X — A) and g/(X — A) are properly [L]-homotopic (in the sense of Definition 2.9) inY — B. Now consider the full subcategory A of the category SHAPE[L] whose objects are Z-sets of the compactum /x^^^. Also consider the category B whose objects are complements of Z-sets in /x^^^ and whose morphisms are weak proper [L]-homotopy classes of proper maps. 2.30. There is a categorical isomorphism T:A->B ^[^] _ xfor each X e A. PROPOSITION

such that T(X) =

Complements of Z-sets in /x^^^ are /x^^^-manifolds (under the assumption of Conjecture 2.7) which have /x^^^-manifold compactifications with "boundaries" (/x^^^ being the compactification and Z-sets being boundaries). The following question has its roots in [28,38,39,50,57,131]. PROBLEM 2.7. When is it possible, for a given /x^^^-manifold X, to find a /x^^^-manifold compactification X such the complement (i.e., "boundary") X — X is a Z-set in X? When can the boundary X — X be assumed to be a /x^^^-manifold?

The last part of the above question might be closely related to the problem of the recognition of spaces of finite [L]-homotopy type. In order to formulate the question, we need to introduce the class of finitely [L]-homotopy dominated spaces. A space X is said to be [L]-homotopy dominated by a space Y if there are two maps / : X -> 7 and g:Y -> X such that [L]

gf - idx. If there exists an ANE([L])-compactum Y such that X is [L]-homotopy dominated by Y and ed(7) < [L], then we say that X is finitely [L]-homotopy dominated. If X is

Infinite dimensional topology and shape theory

335

[L]-homotopy equivalent to an ANE([L])-compactum Y with ed(7) < [L] then X has a finite [L]-homotopy type. If [L] = [{pt}], then the well-known result of Wall [144] states that there are algebraic obstructions for a finitely homotopy dominated ANE-space to have a finite homotopy type. In the case [L] = [S^], the analog of Wall's obstruction vanishes, and consequently every finitely [5"]-homotopy dominated ANECC^"])-spaces has a finite [5"]-homotopy type [50]. In light of these results, we conjecture the following. CONJECTURE 2.1. Let L be a finite complex. If [L] ^ [5^^], then every finitely [L]-homotopy dominated locally compact ANE([L])-space X with ed(X) < [L] has a finite [L]-homotopy type. If [L] is not comparable with any [5"], then there are finitely [L]-homotopy dominated locally compact ANE([L])-spaces X with ed(Z) ^ [L] which do not have a finite [L]-homotopy type.

In the last case it would be very interesting to find explicit constructions of the corresponding algebraic obstructions.

2.8. Open problems In this subsection we discuss possible avenues of development of infinite dimensional topology modulo a finitely complex. As in the classical cases L = {pt}, L = S^ the main objects of investigation are the above introduced spaces fji^^\ v^^^ and manifolds (shortly /x^^^- and yf^^-manifolds) modeled on these spaces. PROBLEM

2.8. Find direct geometric constructions of the spaces /z^^^ and v^^K

The questions below implicitly follow the so-called Edwards' strategy. This is the sequence of steps which proved to be successful in the cases of /^-, R^- and /x"-manifolds and there is no reason to believe that it will not work in the general situation. Here are its main ingredients. Suppose we have a model space X (in our case the spaces v^^^ and /x^^^) and we would like to develop a theory of Z-manifolds. Suppose also that we already have a plausible conjecture about the topological characterization of X-manifolds. The first step of the strategy is to "resolve" a space 7, satisfying the conditions of the conjecture, via a "nice" resolution map fy : Xy ^^ Y defined on an X-manifold. The second step uses Bing's shrinking criterion (Theorem 2.4) and establishes that the map fy is actually a near-homeomorphism. Of course, there are several other essential (and, as a rule, hard to prove) components of this program hidden in each of the indicated steps. Almost all of them are discussed in the subsequent questions. But going back to the first step, a natural question arises: what are those nice maps? In the case of v^^^- and /x^^^-manifolds such maps have already been introduced - these are approximately [L]-soft maps. Therefore our first questions are not unexpected. CONJECTURE 2.2 (v^^^-manifold resolution). PoHsh A(N)E([L])-spaces are approximately [L]-soft images of y^^^-manifolds.

336

A. Chigogidze

CONJECTURE 2.3 (/JL^^^-manifold resolution). Locally compact A(N)E([L])-spaces are proper approximately [L]-soft images of /x^^^-manifolds. The following two questions correspond to the second step of the Edwards' program. CONJECTURE 2.4 (Near-homeomorphisms of v^^^-manifolds). The class of approximately [L]-soft maps / : v^^ -> X of a v^^^-manifold y^^ into a PoUsh A(N)E([L])-space with the [L]-SDAP and of extension dimension ed(X) = [L] coincides with the class of near-homeomorphisms.

2.5 {Near-homeomorphisms of fi^^^-manifolds). A proper approximately [L]-soft map / : / i ^ ^ ^- X of a /x^^^-manifold v)^^ onto a locally compact A(N)E([L])-space with the [L]-DDP and of extension dimension ed(X) = [L] is a nearhomeomorphism. CONJECTURE

The above conjectures immediately imply topological characterizations of the model spaces. 2.6 (Topological characterization of v^^^-manifolds). A Polish space X is a yf^^-manifold if and only if it has the following properties: (i) X€ANE([L]). (ii) ed(X) = [L]. (iii) X is strongly [L]-universal for Polish spaces or, equivalently, X has the [L]-SDAP. CONJECTURE

CONJECTURE 2.7 (Topological characterization of fi^^^-manifolds). A locally compact space X is a /x^^^-manifolds if and only if it has the following properties: (i)

XGANE([L]).

(ii) ed(Z) = [L]. (iii) X is strongly [L]-universal for (locally) compact spaces or, equivalently, X has the [L]-DDP. Z-set unknotting theorems would apparently play an essential role in attempts to prove the above conjectures. CONJECTURE 2.8 (Global Z-set unknotting). Let Y = {v^^\ /x^^^} and Icih.A^

Bhc

a homeomorphism between Z-sets of a F-manifold X such that h c:^ IA, where IA'-A^^ X denotes the inclusion map. Then there exists a homeomorphism H:X ^^ X such that /i = / / / A a n d / / ^ ^ ^ d x . CONJECTURE 2.9 (Local Z-set unknotting). Let Y = {v^^\ ii^^^] and let U e cov(X) be an open cover of a y-manifold X. Then there exists an open cover V e cov(X) such that the following condition is satisfied: (*) if /i: A -)• 5 is a homeomorphism between Z-sets of X which is V-close to IA, then there exists a homeomorphism H:X ^^ X which is ZY-close to idx and which extends h.

Infinite dimensional topology and shape theory

337

The possibility of classification of /x^^^- and v^^^-manifolds by their [L]-homotopy types seems natural. Standard examples of [L]-homotopy equivalences are approximately [L]-soft maps between ANE([L])-spaces of extension dimension not exceeding [L]. CONJECTURE 2.10 (Homotopy classification of v^^^-manifolds). [L]-homotopy equivalent yt^^-manifolds are homeomorphic. More precisely, an [L]-homotopy equivalence of yf^^-manifolds is [L]-homotopic to a homeomorphism.

The situation in the locally compact case is not expected to be so simple. First of all, we have to consider proper [L]-homotopy types. Secondly, even in the compact case we might face obstructions of a different nature (for instance, in the case L = {pt}, simple homotopies have to be considered; see Section 3.2). CONJECTURE 2.11 {Simple homotopy classification of p\^^-manifolds). Let L be a finite complex such that [L] is not comparable with any [5"]. Then simply [L]-homotopy equivalent /x^^^-manifolds are homeomorphic. Moreover, every simple [L]-homotopy equivalence between /x^^^-manifolds is [L]-homotopic to a homeomorphism.

For a given /x^^^-manifold Z, consider a Z-embedding i.X ^^ X which is properly [L]-homotopic to id^. The complement X — i(X) will be called the [L]-homotopy kernel of X and will be denoted by Ker[/^](X). This space is well defined if global Z-set unknotting holds in X. One would think of Ker[L] {X) as an analog of the product X x [0, 1) in the category of /x^^^- manifolds. It is a simple observation that /x^^^-manifolds with homeomorphic (even [L]-homotopy equivalent) [L]-homotopy kernels have the same [L]-homotopy type. We conjecture the converse. CONJECTURE 2.12 {Homotopy classification of ji^^^-manifolds). [L]-homotopy equivalent /x^^^-manifolds have homeomorphic [L]-homotopy kernels. CONJECTURE 2.13 {Triangulation of v^^^-manifolds). Every y^^^-manifold is triangulable, i.e., it admits an [L]-soft map onto a locally finite polyhedron. CONJECTURE 2.14 {Triangulation of ii^^^-manifolds). Every /x^^^-manifold is triangulable, i.e., it admits a proper approximately [L]-soft (and [L]-invertible) map onto a locally finite polyhedron.

The property of being a v^^^- or /x^^^-manifold is hereditary with respect to open subspaces (assuming that Conjectures 2.6 and 2.7 are true). We conjecture that there are no other yt^l- or /x^^^-manifolds. 2.15 {Open embeddings of v^^^-manifolds). Every y^^^-manifold admits an open embedding into y^^^. CONJECTURE

CONJECTURE 2.16 {Open embeddings of /JL^^^-manifolds). The [L]-homotopy kernel of a /x^^^-manifold admits an open embedding into /x^^^.

338

A. Chigogidze

Perhaps the following question should be asked first. PROBLEM 2.9 (Topological homogeneity). Are the spaces v^^^ and /x^^^ topologically homogeneous? For what L are these spaces topological groups?

Note that if L is not connected, then /x^^^ and v^^\ being the Cantor set and the space of irrationals, are both topological groups. Ideally one would expect that the spaces v^^^ and /x^^^ are related to each other. In the known particular examples v^^^ can be identified with the subspace of the compactum /x^^^ the complement /x^^^ — v^^^ of which is locally [L]-homotopy negligible in /x^^^. The last concept is defined as follows. DEHNITION 2.16 ([139]). A subspace Y of a space X is locally [L]-homotopy negligible if for each open subset U of X the inclusion U — Y ^^ U isan [L]-homotopy equivalence.

Now consider the compactum /x^^^ and fix a metric p on it. Theorem 2.5 allows us to construct a sequence {gn : /x^^^ -^ M^^^} of Z-embeddings satisfying the following conditions: (i) p(id^m.gn)<2-\ (ii) gn+\/8n(f^^^^)=idg^(f^[Lly It turns out that the union

is locally [L]-homotopy negUgible in /x^^^ and its complement /x^^^ — U^^^ is a Pohsh AE([L])-space of extension dimension [L] with the [L]-SDAP. In other words, if Conjecture 2.6 is true, then v^^^ ^ /x^^^ - U^^K This observation suggests an alternative approach (through compactifications) to the solution of Conjecture 2.6 under the assumption that Conjecture 2.7 is true. Namely if a space X, satisfying the conditions of Conjecture 2.6, has, in addition, a compactification X which itself is an ANE([L])-space, ed(X) = [L] and the complement X — X is locally [L]-homotopy negligible, then it can easily be seen that X is strongly [L]-universal for compact spaces. Therefore, under our assumption, X is a /x^^^-manifold. This explains why we ask the following question. PROBLEM 2.10. Let X be a space satisfying the conditions of Conjecture 2.6. Does there exist a /x^^^-manifold compactification X of X such that X — X is locally [L]-homotopy negligible in X?

It should be noted here that this question remains open even in the case L = S". We remark also that in the general theory of ANE([L])-spaces, the class of Polish spaces plays the central role. In part this is because every ANE([L])-space (of extension dimension [L]) has an ANE([L])-completion (of the same extension dimension) with locally [L]-homotopy negligible complement (see [53, Proposition 5.9, Case X = {pt}]). The further reduction from Polish spaces to compact spaces is not always possible to make.

Infinite dimensional topology and shape theory

339

The subset U^^^ of /JL^^^ is actually a Z-skeletoid (or, alternatively, an absorber) in /x^^^. There are several possible approaches (such as Anderson's cap sets [32], BessagaPelczynski's skeletoids [18,19], West's absorbers [147], Geoghegan-Summerhill'spseudoboundaries [98], Bestvina-Mogilski's absorbing sets [22], Baars-Gladdines-van Mill's absorbing systems [11]) to the concept of an absorber (see [14] for a complete discussion of the theory of absorbing sets). Further investigation of properties of the space U^^^ gives rise the following conjecture. CONJECTURE 2.17 (Topological characterization of E^^^). The space E^^^ is topologically the only space satisfying the following conditions: (i) U^^^ is a or-compact AE([L])-space. (ii) ed(i:[^]) = [L]. (iii) U^^^ is strongly [L]-universal with respect to a-compact spaces. (iv) Z^^^ has the [L]-SDAP or, equivalentiy, U^^^ is the countable union of its strong Z-sets.

The spaces /x^^^ v^^^ and Z^^^ are the representatives of the first three classes of the Borel hierarchy. A natural question arises of how to define and investigate strongly [L]-universal (in the corresponding sense) AE([L])-spaces of extension dimension [L] in higher Borel classes (see [14,22] for further details in the case [L] = {pt}).

3. The case L = {pt}: "genuine" infinite dimensional topology In this section we consider the case L = {pt}. We will not discuss any statements presented in Section 2 for the general situation unless there are important additions and comments to be made. We use the terminology adapted earlier, but omit extra notational details. For instance in this section ANE-spaces mean ANE([{pt}])-spaces, soft maps are same as [{pt}]soft maps etc. The following statement [23,81] provides a basic example of absolute extensors. 3.1. If C is a convex subspace of a locally convex linear topological space, then C G AE(X) for each metrizable space X. THEOREM

The local convexity in the above statement is essential. The corresponding example has been constructed in [30]. 3.2 ([109]). Every infinite dimensional compact convex subspace of an arbitrary Frechet space is homeomorphic to the Hilbert cube. THEOREM

THEOREM 3.3 ([71]). The following conditions are equivalent for a nonlocally compact Polish group X: (i) X is an ANE-space (respectively, AE-space). (ii) X is an R^-manifold (respectively, is homeomorphic to R*^).

340

A. Chigogidze

Theorems 3.1 and 3.3 imply the following well known result. COROLLARY 3.1 ([4,107]). Every infinite dimensional separable Frechet space is homeomorphic to R^.

It is easy to observe that Polish (compact) absolute extensors are precisely retracts of R^ (of /^, respectively). The following criterion helps us to recognize ANE-spaces. THEOREM 3.4. A space X is an AlSlE-space if and only if for any open cover hi € cov(X) there exists a countable locally finite simplicial complex K (finite dimensional in case X is finite dimensional) and maps p:X^^ \K\ and ^ : | ^ | ^^ X such that the composition qp is U-homotopic to the identity map idx- In addition, we may assume that the cover p~^ (UQ) refines U, where UQ denotes the cover of\K\ consisting of open stars {with respect to the triangulation of K) of vertices of K.

Each Polish ANE-space has the homotopy type of a locally finite polyhedron [118] (the results presented in Section 3.3 imply much a stronger version of this result). Since an open subspace of an ANE-space is itself an ANE-space^ (Proposition 2.11), we see that any open subspace of a Polish ANE-space has the homotopy type of a locally finite polyhedron. It was shown by Cauty [29] that this is in fact a characterizing property of Polish ANE-spaces. 3.5. The following conditions are equivalent for any Polish space X: (i) X is an ANE-space. (ii) Each open subspace ofX has the homotopy type of a locally finite polyhedron.

THEOREM

3.1. Examples of absolute extensors There are several important examples of infinite dimensional absolute extensors. Obviously, the countable powers / ^ and R^, of the closed and open intervals respectively, are among them. The following two spaces are also essential for further discussion:

^ = \{xk}el2: I

J2^kxkf
and a = [{xk} e h'. all but finitely many xk=0}. Certain topological operations preserve the property of being an absolute extensor. Some of them even improve properties of spaces in this sense. ^ The converse of this statement is also true [100].

Infinite dimensional topology and shape theory

341

3.1.1. Cones. Recall that the cone of a space X (notation Cone(Z)) is the set X x [0,1) U {v} topologized as follows. X x [0,1) has the usual product topology and basic open sets containing the vertex are of the form X x (t, 1) U {v} for some t e [0,1). In other words, Cone(X) = (X x [0,1]) U^r {1}, where TT : X x [0,1] ^ [0,1] denotes the projection. Cone(X) is Polish for every Polish space X. 3.1. The following conditions are equivalent: (i) X is an ANE-space. (ii) Cone(Z) is an AE-space.

PROPOSITION

THEOREM 3.6.

Cone(/^) ^ / ^ .

3.1.2. Hyperspaces. The hyperspace exp X of a space X is the set of all non-empty compact subsets of X endowed with the Vietoris topology. Basic open sets ( G i , . . . , G/:) of this topology are sets of the form

I

F G expX: F c ( J G/ and Fr\Gi^9i

for each i =

\,...,k

i=\

If yO is a metric on Z, then the Hausdorff metric PH{F,

T)

= mf{s > 0: F c Bp(T, e) and T c Bp{F, s)}

generates the same topology on exp X. Consider also the closed subspace exp^ X of exp X consisting of all subcontinua of X. 3.7 ([153]). The following conditions are equivalent for any continuum X: (i) X is locally connected. (ii) exp X is an AE-compactum. (iii) exp*^ X is an AE-compactum.

THEOREM

COROLLARY

3.2. If X is an ANE-compactum, then expX and exp'^X also are ANE-

compacta. THEOREM

3.8 ([64]). Let X be a nondegenerate locally connected continuum. Then:

(i) e x p X ^ / ^ . (ii) exp^ X ^ I^ if and only ifX does not contain free arcs. 3.9 ([63]). The following conditions are equivalent for any space X: (i) QxpX^R"^. (ii) X is a connected, locally connected and nowhere locally compact Polish space.

THEOREM

THEOREM 3.10 ([68]). Let n^\ and G be an abelian group. Then the subspaces {F e exp /^: dim F > «} and {F G exp /^: dime F ^ n] o/exp / ^ are homeomorphic to U.

342

A. Chigogidze

COROLLARY 3.3 ([68]). The subspaces {F e exp/^: dimF ^ oo} and {F e exp/^: diniG F > oo} oftxpl^ are homeomorphic to Z^.

Let ANE(Z) denote the subspace of exp X consisting of ANE-compacta. THEOREM

3.11 ([69]). Ifn ^ 3, then ANE(/?") is homeomorphic to Q3.

A related result for n = 2 was obtained in [31]. The space ^ 3 in the above statement is discussed in Section 3.4.1. 3.1.3. Spaces of measures. For a space Z, let CB(C) denote the Banach space of all bounded continuous real-valued functions on X with the sup-norm and C^(Z) be its dual with the weak*-topology. The space C^(X) can be identified [1,143], by integration, with the space of signed, bounded, finitely additive, regular measures defined on the smallest algebra U(X) containing all closed subsets of X. Let A(X) denote the set of functionals in C^(Z) corresponding to a-smooth measures on U(X). Since every asmooth measure on U(X) can be uniquely extended to a Borel measure on X (i.e., a signed, bounded, a-additive measure defined on the smallest a-algebra B(X) containing all closed subsets of X), the linear subspace A{X) c C^(X) can be identified with the space M(X) of Borel (automatically regular in the described situation) measures on X. Further, M~^(X) stands for the subspace of M(X) consisting of nonnegative measures, and P(X) = {/x G M~^(X): /x(X) = 1} is called the space of probability measures. Clearly, for a compactum X, P(X) is either an n-dimensional cube (if |X| = « + 1) or, by [109], is homeomorphic to the Hilbert cube I^ (an easier conclusion that P(X) is an AE-compactum follows from Theorem 3.1). More precisely: 3.12 ([70,91]). The following conditions are equivalent for any space X: (i) X is a compactum with \X\ ^ co. (ii) P{X) is homeomorphic to I^. (iii) M~^(X) is homeomorphic to I^ x [0, 1).

THEOREM

3.13 ([70]). The following conditions are equivalent for any space X: (i) X is a noncompact Polish space. (ii) P{X) is homeomorphic to R^. (iii) M~^(X) is homeomorphic to R^.

THEOREM

One can associate with a space X various types of measures. For instance, let M^(X) c M^(X) be the space of measures with compact support and M'^(X) c M^(X) be the space of measures with finite support. Also, let PK(X) = M^(X) fl P(X) and PF(X) = M+(Z)nP(X). 3.14 ([70]). The following conditions are equivalent for any space X: (i) X is the countable union of finite dimensional compacta. (ii) PF(^) is homeomorphic to a. (iii) M^(X) is homeomorphic to a.

THEOREM

Infinite dimensional topology and shape theory

343

3.15 ([13,70]). The following conditions are equivalent for any space X: (i) X is noncompact, nondiscrete and locally compact. (ii) PK(X) is homeomorphic to U. (iii) M^(X) is homeomorphic to Z.

THEOREM

3.1.4. Spaces of measurable functions. For a space X a function / : [0,1] -> X is said to be measurable if inverse images of open subsets of X are Borel subsets of [0,1]. If / , g:[0,l]^^ X are measurable, then they are equivalent if the set {t e [0,1]: f(t) / g(t)} has Lebesgue measure zero. The set of equivalence classes of measurable functions of [0,1] into X, denoted by Mx, is topologized by the metric

Pig,g)^jf\pif(t),git))fdt, where p is a bounded metric generating the topology of X. 3.16 ([19]). The following conditions are equivalent for any space X: (i) X is a Polish space containing at least two points. (ii) Mx is homeomorphic to R^.

THEOREM

3.1.5. Spaces of autohomeomorphisms. The space of autohomeomorphisms Auth(X) (equipped with the compact-open topology) of a compactum X is a standard source of infinite dimensional manifolds. A comprehensive discussion of related results can be found in [96], [149]. Here we mention only few results. 3.17 ([113]). Let X be a compact n-manifold, n = 1,2. Then Auth(X) is an R^-manifold.

THEOREM

For « > 3, the validity of the above statement is an open question. THEOREM

3.18 ([93,138]). IfX is an I"^-manifold, then Auth(X) is an R"^-manifold.

THEOREM 3.19 ([97,108]). The subgroup Auth^^(X) o/Auth(X) consisting of PL autohomeomorphisms of a PL n-manifold X is a a-manifold.

3.1.6. Spaces of continuous functions. For a countable regular space X, let Cp{X) denote the space of all continuous real-valued functions on X endowed with the topology of pointwise convergence. The topological and linear types of the spaces Cp{X) have been studied extensively. It was shown in [66] that if X is nondiscrete and if Cp {X) is an absolute Ffx^-set, then Cp{X) is homeomorphic to Q2 = ^^ (see Section 3.4.1). 3.2. Topology of I^ -manifolds In this section we present all major results of /^-manifold theory. We begin with the characterization of the Hilbert cube manifolds which was first obtained by H. Torunczyk.

344

A. Chigogidze

THEOREM 3.20 (Characterization of /^-manifolds, [140]). The following conditions are equivalent for a locally compact ANE-space X: (i) X is an I ^-manifold. (ii) X has the disjoint n-disk property for each n eco, i.e., the set

{/6C(/re/2",x):/(/j')n/(/«) = 0} is dense in the space C{I^ ^H^.X) for each n eco. (iii) The set

{f € c(/r e i^, X): f{ir) n f{i^) = 0} is dense in the space C(I^ 0 /^, X). (iv) The set [f eC(Y,X):

f is an embedding]

is dense in the space C{Y, X) for each compact space Y. (v) The set [f eC{Y,X):

f is a Z-embedding}

is dense in the space C(Y, X) for each compact space Y. (vi) X is strongly universal for (locally) compact spaces (= strongly [{Tpi]]-universal for (locally) compact spaces), i.e., for any (locally) compact space B, its closed subspace A, any open cover U G cov(X) and for any (proper) map f: B ^ X such that the restriction f/A is a Z-embedding, there is a Z-embedding g: B -^ X which is U-close to f and such that g/A = f/A. In particular, we obtain a topological characterization of the Hilbert cube. THEOREM 3.21 (Characterization of /^, [140]). The following conditions are equivalent for a compact AE-space X: (i) X is homeomorphic to I^. (ii) X has the disjoint n-disk property for each n eco. (iii) The set

{f e c(/r e i^, X): /(/[") n f{i^) = 0} is dense in the space C(/j^ 0 /|^, X). (iv) The set [f eC(Y,X):

f is an embedding]

is dense in the space C(Y, X) for each compact space Y.

Infinite dimensional topology and shape theory

345

(v) The set {f eC(Y,X):

f is a Z-embedding}

is dense in the space C{Y,X) for each compact space Y. (vi) X is strongly universal for compact spaces, i.e., for any compact space B, its closed subspace A, any open cover U G cov(X) and for any map f : B ^^ X such that the restriction f/A is a Z-embedding, there is a Z-embedding g.B-^X which is U-close to f and such that g/A = f/A. As in the case of i?^-manifolds, the characterization theorem has several corollaries. 3.22 (ANE-theorem, [89]). The following conditions are equivalent for any (locally) compact space X: (i) X is an A(N)E-space. (ii) X X I^ is an I^-manifold (respectively, is homeomorphic to I^).

THEOREM

EXAMPLE

3.1. There is [117] a space X such that X x X ^ I"^ butX ^ I"^.

THEOREM 3.23 ([146]). A countable infinite product of nondegenerate AE-compacta is homeomorphic to I^.

Since / ^ is an AE-space, the projection X x I^ -^ X is a. proper soft map. In particular, it is a fine homotopy equivalence (i.e., approximately soft, or, equivalently, cell-like map). COROLLARY 3.4 ([89,148]). Every locally compact ANE-space is a proper soft image of an I ^-manifold. THEOREM 3.24 (Classification by simple homotopy type, [36]). Simply homotopy equivalent compact I^-manifolds are homeomorphic. Moreover, every simple homotopy equivalence of compact I ^-manifolds is homotopic to a homeomorphism. REMARK 3.1. A similar statement (in terms of proper infinite simple homotopy equivalences) is true for noncompact /^-manifolds [36,132]. THEOREM 3.25 (Classification by homotopy type, [33,36]). I^-manifolds X and Y are homotopy equivalent if and only if the products X x [0, 1) and Y x [0,1) are homeomorphic. Moreover, for every homotopy equivalence f :X ^^ Y the map / x id: Z x [0,1) -> y X [0,1) is homotopic to a homeomorphism. COROLLARY

3.5. I^ is the only contractible compact I^-manifold.

THEOREM 3.26 ([35]). Near-homeomorphisms of I^-manifolds are precisely approximately soft maps.

346

A. Chigogidze

THEOREM 3.27 (Triangulation of /^-manifolds, [36,132]). Every I"^-manifolds is triangulable, i.e., can be represented as the product K x I^ for some locally finite polyhedron K. Moreover, if an I^-manifold X is a Z-set of an I^-manifold Y, then the pair (X, Y) is triangulable, i.e., Y = K x I^ and X = P x I^ where P is a closed subpolyhedron of a polyhedron K.

Observe that Theorems 3.22 and 3.27 imply the following result of J. West [148]. COROLLARY

3.6. Every compact ANE-space is homotopy equivalent to a finite polyhe-

dron. 3.28 (Stability of /^-manifolds, [10]). IfX is an I"^-manifold, then X x / ^ is homeomorphic to X. Moreover, the projection X x I^ -^ X is a near-homeomorphism.

THEOREM

Observe that X ^ X x / ^ ^ X x / ^ x [0,1] ^ X x [0,1]. 3.29 (Open embedding theorem, [36]). For any I^-manifold X the product X X [0,1) admits an open embedding into I^.

THEOREM

THEOREM 3.30 (Z-set unknotting, [7]). Let X be an I^-manifold, A be a locally compact space and F: A x [0, I] -> X be a proper map such that FQ and F\ are Z-embeddings. Then there exists an isotopy H :X x [ 0 , l ] - > X such that HQ = idx and H\FQ = F\. Moreover, if F is a U-homotopy (U € cov(X)), then we may choose H to also be a Uisotopy.

Almost all results discussed above can be generalized to bundles with /^-fibers [142]. In particular, we have the parametric version of the characterization theorem of I^. 3.31 ([142]). The following conditions are equivalent for a proper soft map f :X -^ Y between locally compact ANE-spaces: (i) / is a trivial bundle with fibers I^, i.e., f is topologically the projection X x I^

THEOREM

(ii) Any proper map g: B -^ X of a locally compact space B into X can be approximated by Z-embeddingsh:B ^> X with fh = fg. 3.7. Let / : X ^- Y be a proper soft map of locally compact ANE-spaces. Then the composition fjix'-Xx / ^ - ^ X - > Y is a trivial bundle.

COROLLARY

3.3. Topology of R^-manifolds In this section we present all major results of /?^-manifold theory. We begin with the fundamental result of H. Torunczyk. THEOREM 3.32 (Characterization of 7?^-manifolds, [141]). The following conditions are equivalent for a Polish ANE-space X:

Infinite dimensional topology and shape theory

347

(i) X is an R^-manifold. (ii) The set f e C ' ( ^ { / " : n eo)},Xj:

collection {/(/"): n eco} is discrete in X \

is dense in the space C ( 0 { / " \n eco},X). (iii) X has the strong discrete approximation property, i.e., the set | / ^ ^ ( ® { C * ne(o\,Xy.

collection [f [l^y n e co] is discrete in x\

is dense in the space C ( 0 { / ^ : n ECO], X), where I^ is a copy of the Hilbert cube. (iv) The set [f

GC(Y,X):

f is a closed embedding]

is dense in the space C{Y,X) for each Polish space Y. (v) The set {/ € C(7, X): f is a Z-embedding] is dense in the space C{Y,X) for each Polish space Y. (vi) X is strongly universal for Polish spaces (= strongly [[^i}yuniversal for Polish spaces), i.e., for any Polish space B, its closed subspace A, any open cover U e cov(X) and for any map f: B ^^ X such that the restriction f/A is a Z-embedding, there is a Z-embedding g.B^^X which is U-close to f and such that g/A =

f/A. In particular, we obtain a topological characterization of the space R^. THEOREM 3.33 (Characterization of R^, [141]). The following conditions are equivalent for a Polish AE-space X: (i) X is homeomorphic to R^. (ii) The set

I / ^ ^ ( ® U^* nea)],Xy.

collection {/(/"): n eco] is discrete in X \

is dense in the space C{@{P: n EO)}, X). (iii) X has the strong discrete approximation property. (iv) The set [f eC{Y,X)'.

f is a closed embedding]

is dense in the space C{Y,X) for each Polish space Y.

348

A. Chigogidze

(v) The set {feC{Y,X)\

f isa Z-embedding}

is dense in the space C(Y, X) for each Polish space Y. (vi) X is strongly universal for Polish spaces. These are extremely powerful results, helpful in variety of concrete situations. Some of their immediate corollaries are very hard to prove directly. Observing that the product X X R^ always has the discrete approximation property, we get the following statement. THEOREM 3.34 (ANE-theorem, [137]). The following conditions are equivalent for any Polish space X: (i) X is an A(N)E-space, (ii) X X R^ is an R^-manifold (respectively, is homeomorphic to R^). EXAMPLE

3.2 ([8]). There is a space X such that X xX^R'^butX^

R"^.

3.35 ([141]). A countable infinite product of noncompact Polish AE-spaces is homeomorphic to R^. THEOREM

Since R^ is an AE-space, the projection X x /?^ -> X is a soft map. In particular, it is a fine homotopy equivalence (i.e., approximately soft). THEOREM

3.36. Every Polish ANE-space is a soft image of an R^-manifold.

THEOREM 3.37 (Homotopy classification, [106]). Homotopy equivalent R^-manifolds are homeomorphic. Moreover, every homotopy equivalence between R^-manifolds is homotopic to a homeomorphism. COROLLARY

3.8. R"^ is the only contractible R"^-manifold.

3.38 ([141]). Near-homeomorphisms of R^-manifolds are precisely approximately soft maps. THEOREM

3.9 (Stability of /^^-manifolds, [10,129]). If X is an R"^-manifold, then X X R^ is homeomorphic to X. Moreover, the projection X x R^ -> X is a nearhomeomorphism. COROLLARY

Observe that X ^ X x /?^ ^ X x /?^ x [0,1] ^ X x [0,1]. COROLLARY 3.10. If A isa Z^ -set in an R^-manifold X, then X-Ais Moreover, the inclusion map X — A^^ X is a near-homeomorphism.

an R^-manifold.

Infinite dimensional topology and shape theory

349

THEOREM 3.39 (Triangulation of i?*^-manifolds, [105]). Every R^-manifold is triangulable, i.e., can be represented as the product K x R^ for some locally finite polyhedron K. Moreover, if an R^-manifold X is a Z-set of an R^-manifold Y, then the pair (X, Y) is triangulable, i.e., Y = K x R^ and X = P x R^ where P is a closed subpolyhedron of a polyhedron K. THEOREM 3.40 (Open embedding theorem, [104,105]). R^-manifolds are precisely open sub spaces ofR^. THEOREM 3.41 (Z-set unknotting, [9]). Let X be an R^-manifold, Abe a Polish space, and F \ Ax[0,\\^ X be a map such that FQ and F\ are Z-embeddings. Then there exists an isotopy H :X x [0,1]-^ X such that HQ = idx and H\FQ = F\. Moreover, if F is an U-homotopy (U G cov(X)), then we may choose H to also be a U-isotopy.

It is possible to characterize trivial bundles with fibers R^. 3.42 (Toruiiczyk, West). The following conditions are equivalent for a soft map f :X -^ Y between Polish ANE-spaces: (i) / is a trivial bundle with fibers R^, i.e., f is topologically the projection X x R^ THEOREM

(ii) Any map g'.B^^Xofa Polish space B into X can be approximated by Z-embeddings h'.B^>X with fh = fg. COROLLARY 3.11. Let f :X ^^ Y be a soft map of Polish ANE-spaces. Then the composition fjtx '.X X R^ ^^ X ^^ Y is a trivial bundle.

3.4. Topology of U-manifolds In this section we present results about H- and a-manifolds. In many respect they are similar to those from Sections 3.3 and 3.2. THEOREM 3.43 (Characterization of i:-manifolds, [22,120]). An ANE-space X is a Hmanifold if and only if the following conditions are satisfied: (i) X is a countable union of compacta. (ii) X is a countable union of strong Z-sets. (iii) X is strongly universal for the class of countable unions of compacta, i.e., for any space B which can be written as a countable union of compacta, its closed subspace A, any open cover U G cov(X) and any map f: B ^^ X, such that the restriction f/A is a Z-embedding, there is a Z-embedding g:B —> X which is U-close to f and such that g/A = f/A.

In particular, we obtain a topological characterization of the space U. 3.44 (Characterization of U, [22,120]). An AE-space X is homeomorphic to E if and only if the following conditions are satisfied: (i) X is a countable union of compacta.

THEOREM

350

A. Chigogidze

(ii) X is a countable union of strong Z-sets. (iii) X is strongly universal for the class of countable unions of compacta, i.e., for any space B which can be written as a countable union of compacta, its closed subspace A, any open cover U € cov(X) and any map f .B -^ X, such that the restriction flA is a Z-embedding, there is a Z-embedding g\B -> X which is lA-close to f and such that g/A = f/A. We list some of the corollaries of the characterization theorems. THEOREM 3.45 (ANE-theorem, [89]). The following conditions are equivalent for any space X: (i) X is an A(N)E-space which is a countable union of compacta. (ii) X X E is a Z-manifold (respectively, is homeomorphic to U). THEOREM 3.46 (Classification by homotopy type, [32]). Homotopy equivalent E-manifolds are homeomorphic. COROLLARY THEOREM

3.12. E is the only contractible compact U-manifold.

3.47. Near-homeomorphisms of E -manifolds are precisely approximately soft

maps. 3.48 (Triangulationof 17-manifolds, [32]). Every E-manifoldis triangulable, i.e., can be represented as the product K x E for some locally finite polyhedron K.

THEOREM

3.49 (Stability of T-manifolds, [32]). IfX is a E-manifold, then X, X x X X I^ and X x E are homeomorphic.

THEOREM

r,

3.50 (Open embedding theorem, [32]). The class of E-manifolds coincides with the class of open sub spaces of E. THEOREM

3.51 (Z-set unknotting, [32]). Let X be a E-manifold. Then for each open cover lA € cov(X) there is an open cover V € cov(X) such that the following condition is satisfied: THEOREM

(*) For any homeomorphism h.A-^B between Z-sets of X which is V-close to the inclusion map A^^ X there is a homeomorphism H :X -> X which is U-close to idx and such that H/A = h. By replacing in the results listed above arbitrary compacta by finite dimensional ones we obtain similar statements for the space a. 3.4.1. Other incomplete manifolds. The spaces /^, R^ and E are representatives of the first three classes of the hierarchy of Borel classes. For a countable ordinal a there exist the absolute additive Borelian class Aa and the absolute multiplicative Borelian class A^c^.

Infinite dimensional topology and shape theory

351

Note that Ao = id; Mo consists of all compacta; A\ of all a-compact spaces; M\ is the collection of all Polish spaces, etc. The concept of strong universality which was exploited in Theorems 3.20, 3.32 and 3.43 can be naturally adapted to each of the above classes. It was shown in [22] that for each a ^ 1, there exists an AE-space Qa which is strongly universal with respect to spaces from the class A^Q,. Similarly, for each a ^ 1 there exists an AE-space A^ which is strongly universal with respect to spaces from the class Aa (in this notation Z = A\). These spaces can be characterized topologically as follows. THEOREM 3.52 ([22]). Let a^ \ and X be an AE-space. Then the following conditions are equivalent: (i) X is homeomorphic to Qa {respectively, X ^ Aa). (ii) X = U/Si ^i' ^here Xt G Ma {respectively, Xt ^ Aa) and Xi is a strong Z-set in X for each i. (iii) X is strongly universal with respect to spaces from the class Ma {respectively, from the class Aa).

3.5. Shape theory The shape category SHAPE was defined in Section 2.7 (take L = {pt}). There are several important concepts in shape theory (such as strong shapes, algebraic invariants of shape theory, etc.) which are not even discussed here. We refer to [25,84,85,115] and [149] where the reader can find detailed discussions of various topics and many open problems. Here we concentrate on the geometric side of this concept. Chapman's famous result gives its geometric interpretation. 3.53 (Complement theorem, [33]). Let X and Y be Z-sets in the Hilbert cube I^. Then the following conditions are equivalent: (i) X and Y have the same shape. (ii) The complements I^ — X and I^ — Y are homeomorphic. (iii) The complements I^ — X and I^ — Y are weakly properly homotopy equivalent. ^^

THEOREM

The equivalence of conditions (ii) and (iii), in comparison with Remark 3.1 and Theorem 3.25, shows that complements of Z-sets in / ^ are /^-manifolds of very special type. In order to obtain their internal characterization, let us say that an /^-manifold X has a boundary if there exists a compact /^-manifold X such that X C.X and the complement X — X is a Z-set in X. In this situation we say that X — X is a boundary of X in Z. The problem of characterizing of /'^-manifolds with boundaries has its roots in earlier works [28,131] on a similar problem for usual manifolds. For any contractible open subspace W of R^ not homeomorphic io R^ the product W y. I^ produces an example of a contractible /^-manifold without boundary. It turns out [39] that if an /'^-manifold X satisfies certain necessary homotopy theoretic conditions (finite type and tamenes at oo), then there exists an obstruction ^{X) to X having a boundary. P{X) Hes in the quotient 10 See p. 28.

352

A. Chigogidze

Soo(X) of the group S{X) of all infinite simple homotopy types on X by the image of the Whitehead group Wh7i\ (X). This group depends only on the inverse system [7T\(X — K): ^ is a compact subset of Z } . It turns out that the different boundaries that can be put on X constitute a whole shape class. The different ways of putting boundaries on X can be classified by elements of the inverse limit of the above indicated inverse system. The next step in this direction is to ask when /^-manifolds (with boundaries) admit a polyhedral boundary? This problem was considered and solved in [38]. There are additional obstructions to an /^-manifold having a polyhedral boundary. This obstruction can be understood via the following result. THEOREM 3.54 (Proper finiteness obstruction theorem, [38]). If X is an I^-manifold which is finitely dominated near oo, then there is an element cfp(X) in^^

Ko7t\E(X) = \im[Kon\{X — K): K is a compact subset of X} which vanishes if and only ifX has finite type near oo. Moreover, Gp{X) is an invariant of proper homotopy type near oo. Several homotopy theoretical questions for complexes have their counterparts in shape theory. For instance, if a continuum X is shape dominated by a finite complex, is it shape equivalent to a finite complex? As in the case of homotopies, the answer is not always affirmative [90]. There is an intrinsically defined shape theoretic version of Wall's obstruction w(X) G KQ7C\ (X) which vanishes if and only if X is shape equivalent to a finite complex (here fc\ (X) denotes the Cech fundamental group of X).

4. The case L = S^: "n-dimensional" infinite dimensional topology In this section we discuss the case L = S^ which, as was mentioned above, produces the theory of n-dimensional Menger and Nobeling manifolds. Clearly ANE([5"])-spaces are precisely LC"~^-spaces, UV^^''^-msips are called L^V^-maps (not [/V"~^-maps as usual), etc. We recall that [5"]-homotopies and [5'"]-shapes in this paper are called n-homotopies and n-shapes, respectively (instead of the originally used terminology: (n — 1)-homotopies and (n — l)-shapes). Of course, the Menger compactum p.^^ ^ modulo [S"] is denoted, as usual, by /x". The same applies to the Nobeling space modulo [5"]: v" = v^'^^^

4.1. Menger manifolds We begin with the standard construction of the Menger compactum. Let n > 0. We partition the unit cube /^""^^ lying in (2n + 1)-dimensional Euclidean space 7?^"+^ into 3^"+^ ^ ^ Recall that Wall's finiteness obstruction is an element of the reduced projective class group ^o^l (^)-

Infinite dimensional topology and shape theory congruent cubes of the "first rank" by hyperplanes drawn perpendicular to the edges of the cube Z^^+i at points dividing the edges into three equal parts, and we choose from these 32^+1 cubes those which intersect the n-dimensional skeleton of the cube /2"+i. The union of the selected cubes of the "first rank" is denoted by /(n, 1). In an analogous way, we divide every cube entering as a term in /(«, 1) into 3^"+! congruent cu6es of the "second rank", and the union of all analogously selected cubes of the "second rank" is denoted by /(«, 2). If we continue the process we get a decreasing sequence of compacta /(n,l)D/(n,2)2---. The compactum /x" = f^{I(n,i): n = 1,2,...} is called the n-dimensional universal Menger compactum. Note that JJP = Af^"+^ where M^, O^n
353

354

A. Chigogidze

THEOREM 4.1 (Characterization of /x"-manifolds, [20]). The following conditions are equivalent for any n-dimensional locally compact AH^{n)-space X: (i) X is a jji^ -manifold. (ii) X has the disjoint n-disk property, (iii) The set

{/ 6 C(y, Z): f is an embedding^ is dense in the space C{Y, X) for each at most n-dimensional compact space Y. (iii) The set {/ G C(Y, X): f is a Z-embedding] is dense in the space C(F, X) for each at most n-dimensional compact space Y. (iv) X is strongly universal for at most n-dimensional {locally) compact spaces (= strongly [S"]-universal for (locally) compact spaces), i.e., for any at most n-dimensional (locally) compact space B, its closed subspace A, any open cover hi G cov(X) and any (proper) map f: B -^ X such that the restriction f/A is a Z-embedding, there is a Z-embedding g:B ^^ X which is U-close to f and such that g/A = f/A. Additionally, if X is compact and (n — l)-connected (i.e., X e AE([5"]) = AE(n)), then conditions (ii)-(vi) give a topological characterization of the compactum fx". 4.2 (Resolution theorem, [20,72,73]). There are maps /„:/x" ^ I"^ and gn'. /Ji^ -^ /x" satisfying the following properties: (i) The maps fn and gn are n-invertible, (n — \)-soft andpolyhedrally n-soft. (ii) All the fibers of the maps fn and gn are homeomorphic to /x". (iii) The inverse images of ANE(n)-compacta under the maps fn and gn are p.^-manifolds. (iii) The maps fn and gn both satisfy the parametric version of the disjoint n-disk property, that is, any two maps a, P'.P -^ p^ can be arbitrarily closely approximated by maps a', )6': /" —> /x" such that fnOi' = fnCi> fnP' = fnP, gn^^' = gnOt, g^P' = g^P and im(aO O im(j60 = 0. THEOREM

At first glance it is not clear what is the analog of the operation of "taking the product by I^'" in /x"-manifold theory - the operation which is involved in the formulations of the triangulation (Theorem 3.27) and stability (Theorem 3.28) theorems for /^-manifolds. Taking the product X x 7*^ of a space X and the Hilbert cube / ^ may be interpreted as taking the inverse image 7t^\X) of a space X c I^, where 7T\:I^ x I^ -> I^ denotes the natural projection onto the first coordinate. It turns out that the map gn = fn/fn^(l^^): /x" -> /x" in Theorem 4.2 may be thought of as the analog of the projection 7t\ in the theory of /x"-manifolds. If this is agreed, everything then falls in place.

Infinite dimensional topology and shape theory

355

THEOREM 4.3 (Triangulation of /x"-manifolds, [73]). For any ix^-manifold M, there is an n-dimensional polyhedron P such that for any embedding of P into /JL^ the inverse image g~^(P) is homeomorphic to M. THEOREM 4.4 (Stability of /x"-manifolds, [73]). For any /JL""-manifold M in /i", the inverse image g~^ (M) is homeomorphic to M. THEOREM 4.5 (Classification by proper n-homotopy type, [20]). Properly n-homotopic /JL^-manifolds are homeomorphic. Moreover, every proper n-homotopy equivalence of 11^-manifolds is properly n-homotopic to a homeomorphism. THEOREM 4.6 (Classification by n-homotopy type, [48]). Two ix^-manifolds are n-homotopy equivalent if and only if their n-homotopy kemels^^ are homeomorphic. THEOREM 4.7 ([20]). Near-homeomorphisms of /JL^-manifolds are precisely proper approximately n-soft maps (= proper UV^-maps). THEOREM 4.8 (Open embedding theorem, [48]). The n-homotopy kernel of an fi^-manifold admits an open embedding into /x".

4.9 (Local Z-set unknotting, [20]). Let X be a fi^-manifold. Then for each open cover U € cov(X) there is an open cover V G cov(X) such that the following condition is satisfied: THEOREM

(*) For any homeomorphism h\A^>B of Z-sets ofX such that h is V-close to the inclusion map A^^ X, there is a homeomorphism H :X -> X which is U-close to idx and such that H/A = h. 4.10 (Global Z-set unknotting, [20]). Let Xbea ^J^-manifold and leth:A-> B be a homeomorphism between Z-sets of X which is properly n-homotopic to the inclusion map A ^> X. Then there exists a homeomorphism H :X ^^ X which is properly n-homotopic to idx and such that h = H/A. THEOREM

4.1 (Homogeneity of /x'^, [2,3,20]). The universal Menger compactum /x" is topologically homogeneous. COROLLARY

4.2. Nobeling manifolds Much less is known about the n-dimensional Nobeling space v". Let us first recall the definition of the space NJ^ (O^n
356

A. Chigogidze

CONJECTURE 4.1. A Polish space X is homeomorphic to v" provided that X satisfies the following conditions: (i) dimX = n. (ii) XGAE([5"]) = AE(n) = LC"-i nC'-K (iii) X is strongly universal for at most n-dimensional Polish spaces (= strongly [S^]universal for Polish spaces), i.e., for any at most n-dimensional Polish space B, its closed subspace A, any open cover U e cov(X) and for any map f: B ^^ X such that the restriction f/A is a Z-embedding, there is a Z-embedding g.B^^X which is Z^-close to / and such that g/A = f/A.

It was observed in [136] that /?2«+i _ v« is a skeletoid with respect to a certain collection of n-dimensional polyhedra in /?2«+i ^^^^^ |^9g] f^^ ^^ construction of different skeletoids in R^). The compactum /x" also contains a skeletoid 17", which is nothing else but the space Z't'^"^ constructed on p. 33. THEOREM 4.11 ([59]). v" is homeomorphic to the pseudo-interior ji^ — E^ of the Menger compactum /x".

Most of the results listed below are based on the above theorem. 4.2 ([59,65]). Let k^2n interior sj^ of R^.

COROLLARY

COROLLARY

-\- I. Then y" is homeomorphic to the pseudo-

4.3. The space v" is topologically homogeneous.

According to [51], every /x"-manifold M contains a skeletoid E^(M). Their complements v^(M) = M — E^(M) are v"-manifolds of a special type. Let us call them regular v^-manifolds. It is not known whether every y"-manifold is regular. 4.12 ([51]). The following conditions are equivalent for any space X: (i) X is homeomorphic to an open subspace ofv^. (ii) X is a regular y" -manifold.

THEOREM

4.13 (Classification by «-homotopy type, [51]). n-homotopy equivalent regular y^ -manifolds are homeomorphic.

THEOREM

The following corollary of the above theorem deserves to be stated separately (I have no idea how to prove it directly). 4.4 ([51]). An {n — I)-connected open subspace ofv^ is homeomorphic to v^. In particular, a connected open subspace ofv^ is homeomorphic to v^.

COROLLARY

Infinite dimensional topology and shape theory

357

4.3. n-shape theory The «-shape category ^z-SHAPE can be defined by following the general definition (Section 2.7, the case L = 5"). Originally this concept was defined in [43] (compare with [94]). Isomorphic objects of this category are said to have the same n-shape. THEOREM 4.14 (Complement theorem for /x'^, [43]). Let X and Y be Z-sets in the Menger compactum fi". Then the following conditions are equivalent. (i) The complements fi^ — X and /x" — F are homeomorphic. (ii) n — Sh(X) =n — Sh(F), i.e., X and Y have the same n-shape.

All results discussed in Section 3.5 have their natural counterparts in the «-shape theory. We refer the reader to [57] for details. Here we only remark that the natural analogs of algebraic obstructions (see Section 3.5) corresponding to the «-shape category vanish. The explanation of this phenomena is provided in the next statement (see also p. 335) which shows that the analog of Wall's obstruction vanishes in the n-homotopy category. THEOREM 4.15 ([50]). If a locally finite polyhedron is n-homotopy dominated by an at most n-dimensional finite polyhedron, then it is n-homotopy equivalent to an n-dimensional finite polyhedron.

5. Non-metrizable manifolds Non-metrizable manifolds are traditionally outside of what is called Geometric Topology. On the other hand, the uncountable powers P and R^ of the closed segment and of the real line are extremely important and interesting objects from various points of view and deserve to be discussed here. Below we present a brief outline of an extension of the classical ANE-theory (case L = {pt}) to the nonmetrizable case. One of the main differences of a technical nature between the metrizable and nonmetrizable parts of the ANE-theory is that the apparatus of uncountable inverse spectra becomes the main tool of investigation in the nonmetrizable case. We recommend [51] for a comprehensive discussion of related topics. There is another, but not less essential, difference hidden in the classical definition of ANE-spaces. Indeed, according to the definition of the relation X e AE(7) (see Section 2.1) the real line is not an AE-space in the class of Tychonov (i.e., completely regular and Hausdorff) spaces. This fact immediately follows from the simple observation that a nonnormal space always contains a closed subspace and a continuous real-valued function not extendable onto the whole space. In order to modify this definition, let us analyze closely the classical extension problem. The most general form of this problem asks: EP: Does a map / : F ^- P, defined on a (closed) subspace 7 of a space X, admit an extension f :X -^ P

358

A. Chigogidze

^P Consider the category TYCH of Tychonov spaces and continuous maps on the one hand, and the category VECT of vector spaces and linear maps on the other. There is a natural contravariant functor C: TYCH -^ VECT assigning to each space X the vector space C(X) of all continuous real-valued functions on X, and to each continuous map / : X -> F the induced linear operator C ( / ) :€{¥)-> C{X). Applying the functor C to the above Extension Problem (see [134, Theorem 1.1.10]) we arrive at the corresponding Lifting Problem in the category VECT: LP: Does the map C(f): C(P) -> €(¥) have a lifting a : C(P) exist a map a such that C(f) = C{i)(x)l

C{X) (i.e., does there

C{X) C{i)

C(Y)-

C(f)

C(P)

The existence of such a lifting produces the following necessary condition for EP io have a solution. NC: C(/)(C(P)) = C(i)(a(C(P)) c C(i)(C(X)) = C(X)/Y, where C(X)/Y C(Y): there exists cp e C(X) such that cp/Y = (p}.

= {(p e

Obviously, by the Tietze-Urysohn Extension Theorem, condition NC is automatically satisfied for any f :Y ^^ P defined on a closed subspace F of a normal space X. Now recall that a subset A of a space X is called functionally open (closed) if there is a function f.X^R such that A = f~\R{0}) (respectively, A = f~\0)). Let us call a functionally open neighborhood f/ of a subset A in a space X stable if there is a functionally closed subset Z of X such that A c Z c f/. Observe that if a subset A is C-embedded in X, then every functionally open neighborhood of A is stable. Based on the above observation we arrive at the following definition. DEFINITION 5.1. Let X and L be Tychonov spaces. We say that L is an absolute (neighborhood) extensor of X, and write L e A(N)E(X), if for each subspace 7 of X any map f'.Y^L such that C{f){C{L)) c C{X)/ Y admits an extension f.X^L (respectively, f.G^^L) over X (respectively, over a stable neighborhood G of A in X).

Note that we have dropped the requirement that Y be closed in X.

Infinite dimensional topology and shape theory

359

This approach allows us to define the extension dimension of any space ^^ and to extend the theory of ANE([L])-spaces to the nonmetrizable case. Realcompact spaces play an important role in this theory. These are spaces homeomorphic to closed subspaces of powers of the real line. Information about realcompact spaces, and in particular about Hewitt realcompactification vX, can be found in [51,99]. 5.1 ([53]). Let X be a (Tychonov) space and L be a countable complex. Then L e AE(X) if and only ifLe AE(i;X).

THEOREM

THEOREM 5.2 ([53]). The following conditions are equivalent for any realcompact space X and a countable complex L: (i) ed(X) ^ [L]. (ii) X is the limit space of a Polish spectrum Sx = [Xa, pa, A} such that ed(Xc^) ^ [L] for each a ^ A.

Theorems 5.1 and 5.2 reduce the theory of extension dimension of general spaces to that of Polish spaces. Definition 2.2 can also be extended to the nonmetrizable case. DEFINITION 5.2. Let L be a complex. A space X is called an absolute (neighborhood) extensor modulo [L] (shortly, A(N)E([L])-space) if for each space Z with ed(Z) < [L] and each subspace ZQ of Z, any map / : ZQ ^^ X, such that C(/)(C(X)) c C(Z)/Zo, can be extended to (some stable functionally open neighborhood of ZQ) Z.

Since [L] ^ [5^], every ANE([L])-space is an AE([5^])-space (i.e., AE(0)-space in the terminology of [51]). Therefore ANE([L])-spaces have all the properties of AE(0)spaces discussed in [51]. In particular, such spaces are realcompact. Within the class of realcompact spaces. Definition 2.3 admits the following extension. DEFINITION 5.3. Let L be a complex. A map f :X ^^ Y of a realcompact spaces is said to be [L]-soft if for each realcompact space Z with ed(Z) ^ [L], for its (closed) subspace ZQ, and for any two maps g:Zo^^ X and h:Z ^^ Y such that fg = h/Zo and C(g)(CiX)) c C(Z)/Zo, there exists a map k:Z^ X such that JC/ZQ = g and fk = h. Obviously the class of AE([L])-spaces coincides with the class of realcompact spaces, the constant maps of which are [L]-soft. Using techniques of [51,53] (see also related papers [41,42]) one can prove the following statements. 5.1. Let L be a complex and r be an infinite cardinal. Then there is an [Ly soft map /[L],r * ^[L],r -^ ^^ satisfying the following conditions'. (i) ed(X[L],r) = [LI '

PROPOSITION

(ii) M;(X[L],T) = T. ^^ In this section all spaces are Tychonov.

360

A. Chigogidze

The above proposition is a key ingredient of the proof of the following statement which characterizes nonmetrizable A(N)E([L])-spaces. 5.3. Let L be a complex. A space X of weight x ^ co is an A.(N)E{{LY^-space if and only if it is the limit space of an inverse spectrum «Sx = {^a, p^"^^ t} satisfying the following conditions'. (i) XQ is a Polish A(H)E{[LY)-space. (ii) Each short projection p'^^^ : ^a+i -> Xa is [L]-sofi. (iii) Each short projection p^'^^ has a Polish kernel in the sense of [51]. (iv) The spectrum Sx is continuous, i.e., for each limit ordinal a
For [L] = [S^] and compact X, the above result was proved in [103]. For [L] = [5"] (with n ^ 1) and compact X, it was obtained in [72] (see also [92] and [121]). The case [L] = [{pt}] and compact X was considered in [126]. All these particular results were generalized for noncompact spaces in [41]. Theorem 5.3 has various applications. Proofs of most of the results below are based on this statement. Here is another, sometimes more convenient, version of the above statement. THEOREM 5.4. Let L be a complex and r ^ co. A space X is an A(N)E([L])-space if and only if it is the limit space of a r-spectrum Sx = {Xa, Pa^T^} satisfying the following conditions: (i) Xa is an A(N)E([L])-space of weight r, a e A. (ii) Each limit projection pa'.X -^ Xa is [L]-soft.

Theorem 5.3 and Proposition 2.17 imply the following result. 5.5 ([61]). Let L be a noncontractible connected complex and X be an ANE([L])-compactum. Ifcd(X) ^ [L], then X is metrizable. THEOREM

The case [L] = [5"] was earlier proved by Dranishnikov [72]. Theorem 5.5 strengthens also the result of Scepin [127] on metrizability of finite-dimensional ANE-compacta. Note that unlike the compact case, there exist noncompact AE([L])-spaces of extension dimension [L] and of arbitrary weight.

5.1. r - limitation topology Let X and Y be arbitrary Tychonov spaces and r be an arbitrary infinite cardinal number. We are going to introduce a topology (r-limitation topology, depending on r) on the set C{Y,X) of all continuous maps from Y into X. The space thus obtained will be denoted by Cr (Y, X). We recall that cov(X) denotes the collection of all countable functionally open covers of the space X. For each map / : 7 ^^ X the sets of the form B{f, {Ut\ teT})

= [ge C(F, X): g is ZY^-close to / for each t

eT],

Infinite dimensional topology and shape theory where | r | < r and Ut e cov(X) for each t eT, arc declared to be open basic neighborhoods of the point / in Cr (Y, X). The maps contained in the neighborhood B(f, {Ut: t eT}) are called {Ut: t e r}-close to/. If a space X has a countable basis, then, obviously, the space C^(y, X) (for any space Y) coincides with the space C(F, X), endowed with the limitation topology introduced in Section 2.4.1. Several common concepts in the metrizable case have to be modified for the nonmetrizable case. For instance, we call a closed subset A of a space X a Zr-set if the set { / G C r ( X , Z ) : / ( X ) n A = 0} is dense in the space Cx {X, X). Similarly, elements of the closure (in Cx {X, Y)) of the set of homeomorphisms of X onto Y are called T-near-homeomorphisms . Interestingly enough, it is much easier to detect Z^-sets in the nonmetrizable case. For instance, every metrizable or finite-dimensional compactum in the Tychonov cube /^ is a Z^-set [60].

5.2. P-manifolds In this section we assume that r is an uncountable cardinal. A Lindeloff space X is an r -manifold if it has a countable functionally open cover, each element of which is homeomorphic to a functionally open subspace of the Tychonov cube P. A characterization of /^-manifolds was obtained by Scepin [128]. First we need to extend Definition 2.5 to the nonmetrizable case. DEFINITION 5.4. For a given complex L we say that a (locally) compact space X is strongly T-[L]-universalfor compact spaces if for any (locally) compact space B such that ed(5) ^ [L] and w(B) < r, its closed subspace A, any (proper) map f : B ^^ X such that the restriction f/A is a Zr-embedding, and any neighborhood U of f in Cx(B, Z), there is a Zr-embedding g eU such that g/A = f/A. THEOREM 5.6 (Characterization of /^-manifolds, [51,128]). The following conditions are equivalent for a locally compact and Lindeloff I^^-space X of weight r: (i) X is an P -manifold. (ii) X is strongly r-[{pt}]-universal for compact spaces. (iii) For each compact space B of weight < r, the set of embeddings is dense in the space Cx(B, X). (iv) The set of embeddings is dense in the space Cx ({0,1}, X). (v) X is homogeneous with respect to the pseudo-character, i.e., (p{x, X) = r for each point X e X.

Condition (iii) in this theorem is the main difference between the metrizable and nonmetrizable cases. It implies statements which are not even true for r =co. For instance, if

361

362

A. Chigogidze

the product of two compact spaces is homeomorphic to /^, then one of them itself must be a copy of /^. Compare with Example 3.1. Another strange fact: the cone Cone(/^) is not homeomorphic to /^ (the vertex of the cone is a G5-point violating condition (v)), whereas Cone(/^) ^ / ^ (Theorem 3.6). The following statement provides a natural bridge between the metrizable and nonmetrizable cases. 5.7 ([128]). The following conditions are equivalent for any space X: (i) X is an P -manifold. (ii) There is an I ^-manifold (unique up to a homeomorphism) Y such that X ^ Y x I^.

THEOREM

This theorem has several corollaries. THEOREM

5.8 (Stability of /"^-manifolds, [128]). If X is an P-manifold, then X ^

Xxr. 5.9 (Triangulation of/^-manifolds, [128]). Every I'^-manifold is triangulable, i.e., can be represented as the product K x P for some locally compact polyhedron K.

THEOREM

5.10 (Open embedding theorem, [51]). For any P-manifold X, the product X X [0, 1) is homeomorphic to a functionally open subspace of P.

THEOREM

THEOREM 5.11 (Homotopy classification, [128]). P-manifolds X and Y are homotopy equivalent if and only if the products X x [0, 1) and F x [0, 1) are homeomorphic. THEOREM 5.12 (ANE-theorem, [128]). The product X x P is an P-manifold if and only ifX is a locally compact and Lindeloff ANE-space. THEOREM 5.13 (Global Z^-set unknotting, [51,60]). Suppose that the homeomorphism g'.A^^B between Z^-sets of the compact P -manifold X is homotopic to the inclusion map A"^^ X (r > oj). Then there is an autohomeomorphism G.X ^^ X which extends g and is homotopic to id^.

5.14 (Local Zr-set unknotting, [51,60]). Let X be a compact P-manifold, r > CO. For each collection {Ut'. t eT}c. cov(X), | r | < r, there exists another collection {V/i t G T'}, | r ' | < T, such that the following condition is satisfied:

THEOREM

(*) For any homeomorphism h.A^^B between arbitrary Zx-sets of X which is [Vt\ t e T'}-close to the inclusion map A^^^ X, there is a homeomorphism H.X ^^ X such that H/Z = h and which is {Ut'. t e T]-close to idx5.15 ([51]). Each compactum of weight r, admitting a soft map onto the Tychonov cube P, is homeomorphic to P. THEOREM

Infinite dimensional topology and shape theory It is not hard to see that the group A^ acts on the Hilbert cube /^, fixing a single point and acting freely off that point [88]. Hence A^ acts freely on the noncompact /^-manifold I^ — {pt} ^ I^ X [0,1). Are there compact /'^-manifolds admitting a free action of the group A^? This question (R.D. Edwards [88]) is a weaker version of the wellknown Hilbert-Smith conjecture^^, and to the best of my knowledge still remains open. The following statement (the proof of which is based on the spectral technique) answers a nonmetrizable version of Edwards' question. 5.16 ([51]). There is no nonmetrizable ANE-compactum admitting a nontrivial semi-free action of any nonmetrizable compact group.

THEOREM

5.3. R^-manifolds A space Z is an /?^ -manifold if it has a countable functionally open cover each element of which is homeomorphic io R^. A characterization of /?^-manifolds was obtained in [41]. We start with the nonmetrizable counterpart of Definition 2.8. DEFINITION 5.5. For a given complex [L] we say that a realcompact space X is strongly r-[L]-universal for realcompact spaces if for any realcompact space B such that ed(5) < [L] and R-w{B) ^ r, its closed subspace A, any map f :B -> X such that the restriction f/A is a Z^-embedding, and any neighborhood U of f in Cx(B, X), there is a Zr-embedding g eU such that g/A = f/A. THEOREM 5.17 (Characterization of /?^-manifolds, [51,128]). The following conditions are equivalent for any ANE-space X of weight r: (i) X is an R^ -manifold. (ii) X is strongly r-[{pi}]-universal for realcompact spaces. (iii) For each realcompact space B of R-weight < r, the set of C-embeddings is dense in the space Cx{B, X). (iv) The set ofembeddings is dense in the space Cj (co,X).

Condition (iii) in this theorem is the main difference between the metrizable and nonmetrizable cases. It implies statements which are not even true for T = (W. For instance, if the product of two spaces is homeomorphic to R^, then one of them must be a copy of R^. Compare with Example 3.2. The following statement provides a natural bridge between the metrizable and nonmetrizable cases. 5.18 ([51]). The following conditions are equivalent for any space X: (i) X is an R^-manifold. (ii) There is an R^-manifold {unique up to a homeomorphism) Y such that X ^ 7 x R^.

THEOREM

The Hilbert-Smith conjecture asks whether a locally compact group acting effectively on an n-manifold must be a Lie group.

363

364

A. Chigogidze

This theorem has several corollaries. THEOREM

5.19 (Stability of Z?''-manifolds, [51]). If X is an R^-manifold, then X ^

X^R\ 5.20 (Triangulation of 7?^-manifolds, [51]). Every R^-manifold is triangulable, i.e.y can be represented as the product K x R^ for some locally compact polyhedron K. THEOREM

THEOREM 5.21 (Open embedding theorem, [51]). The following conditions are equivalent for any space X: (i) X is an R^ -manifold. (ii) X is homeomorphic to a functionally open subspaces of R^. THEOREM 5.22 (Homotopy classification, [51]). Homotopy equivalent R^ -manifolds are homeomorphic.

5.23 (ANE-theorem, [51]). The product X x R^ is an R^ -manifold if and only if X is an ANE-space.

THEOREM

THEOREM 5.24 ([51]). Every space of weight r, admitting a soft map onto R^, is homeomorphic to R^.

We conclude this section with the following generahzation of Corollary 3.1. THEOREM 5.25 ([47]). Let x be an uncountable cardinal. Then the following conditions are equivalent for any locally convex linear topological space X of weight r: (i) X is homeomorphic to R^. (ii) X is an AE-space. (iii) X is an AE(0)-space (recall that AE(0) = AE([5'^])).

6. Topological groups Quite often, simple topological assumptions in the presence of an additional algebraic structure have surprisingly strong consequences. We have seen examples of such situations in Corollary 3.1 and Theorem 5.25. Two more examples are provided in the following statements. PROPOSITION

6.1 (Pontryagin-Haydon, [103,124]). Every compact group is an AE(0)-

space. 6.2 ([17]). The following conditions are equivalentfor a zero-dimensional topological group G: (a) G is topologically equivalent to the product {TJIY X W. (b) G is an AE(S))-space. PROPOSITION

Infinite dimensional topology and shape theory

365

Below we consider compact topological groups from the point of view of the general theory of absolute extensors.

6.1. Abelian case - the Whitehead problem and characterization of Torus groups The problem of the topological characterization of torus groups (i.e., powers of the circle group T) has been of considerable importance in the theory of compact abelian groups. The following statement collects most of the known information. 6.3. Let G be a connected compact abelian group. Then the following conditions are equivalent. (1) G is arcwise connected. (2) G is locally arcwise connected. (3) The map exp^^ : L(G) —> G is surjective. (4) The map exp^^ : L(G) -> G is open. (5) Ext(G, Z) = 0, i.e., the character group GofG is a Whitehead group. (6) Ext(G, Z) = 0 and each pure subgroup ofG of finite rank splits. In addition, if G is metrizable, then the above conditions are equivalent to each of the following: (7) G is free. (8) G is ^\-free, i.e., every countable subgroup ofG is free. (9) G is projective. (10) G is injective {in the category of compact abelian groups). (11) G is locally connected. (12) G is a torus. PROPOSITION

In the nonmetrizable case situation is quite different. For instance, the equivalence (8) <^ (11) remains true, whereas (11)<^(12) does not. The equivalence (5) ^ (7) is undecidable in ZFC [130]. In topological terms this means that: (i) There are (necessarily nonmetrizable) connected and locally connected compact abelian groups which are not tori, (ii) It is undecidable within ZFC that the arcwise connectedness of a nonmetrizable compact abelian group forces the group to be isomorphic to a torus. The following result provides a topological characterization of torus groups. THEOREM 6.1 ([54]). The following conditions are equivalent for a compact abelian group G: (i) G is a torus group (both in the topological and algebraic senses). (ii) G is an AE{1)-compactum. (iii) The exponential map cxpQ : L(G) -> G is 0-soft.

Since, by duality, any statement in the category of compact abelian groups generates an equivalent statement in the category of abelian groups (see Proposition 6.3), it might be worth examining Theorem 6.1 from this point of view. By the equivalence (7) 4^^ (12)

366

A. Chigogidze

in Proposition 6.3, the statement corresponding to Theorem 6.1 in the category of abelian groups must provide a characterization of free groups. Since dim G = rank(G) for a finite dimensional compact abeUan group G, and since the concept of AE(1) is defined in terms of 1-dimensional testing compacta, this would lead us to the concept of 1-projectivness (restrict, in the standard definition of projectivness, domains of testing epimorphisms by groups of rank 1) for abelian groups, and to the hope of a subsequent characterization of free abelian groups in these terms. The so obtained version of AE(1) will differ from the original topological concept of AE(1) and will not be a replacement for the latter in Theorem 6.1. The reason is simple - every 1-dimensional connected compact (abelian) group is metrizable and, consequently, the changed version of AE(1) would essentially coincide with arcwise connectedness. This shows that Theorem 6.1 provides a purely topological characterization of tori.

6.2. Non-abelian case Extension theory allows us to characterize simple, connected and simply connected Lie groups. Here we present corresponding statements. Note that every connected Lie group topologically is an AE(1)-space and every connected and simply connected Lie group is an AE(2)-space. 6.2 ([52]). The following conditions are equivalent for a compact group G: is a simply connected AE(l)-compactum, is an AE(2)-compactum. is an AE(3)-compactum. is a product of simple, connected and simply connected compact Lie groups.

THEOREM

(i) (ii) (iii) (iv)

G G G G

This statement immediately implies COROLLARY 6.1 ([52]). The following conditions are equivalentfor a nontrivial compact group G: (i) G is an AE(2)-group with n^{G) = Z. (ii) G is a simple, connected and simply connected Lie group.

Note that the implication (ii) =>• (i) in Corollary 6.1 is due to R. Bott (see [119, Theorem 3.8]).

References [1] A.D. Aleksandrov, Additive set-functions in abstract spaces. Mat. Sbomik 8 (1940), 307-348; 9 (1941), 563-628; 13 (1943), 169-238. [2] R.D. Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. Math. 67 (1958), 313-324. [3] R.D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. Math. 68 (1958), 1-16.

Infinite dimensional topology and shape theory

367

[4] R.D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 72 (1966), 515-519. [5] R.D. Anderson, On topological infinite deficiency, Michigan Math. J. 126 (1967), 365-383. [6] R.D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals. Trans. Amer. Math. Soc. 126 (1967), 200-216. [7] R.D. Anderson and T. A. Chapman, Extending homeomorphisms to Hilbert cube manifolds. Pacific J. Math. 38 (1971), 281-293. [8] R.D. Anderson, D.W. Curtis and J. van Mill, A fake topological Hilbert space. Trans. Amer. Math. Soc. 272 (1982), 311-321. [9] R.D. Anderson and J.D. McCharen, On extending homeomorphisms to Frechet manifolds, Proc. Amer. Math. Soc. 25 (1970), 283-289. [10] R.D. Anderson and R.M. Schori, Factors of infinite-dimensional manifolds. Trans. Amer. Math. Soc. 142 (1969), 315-330. [11] J. Baars, H. Gladdines and J. van Mill, Absorbing systems in infinite-dimensional manifolds. Topology Appl. 50 (1993), 147-182. [12] T.O. Banakh, Characterization of convex Z-sets in linear metric spaces, Funct. Anal. Appl. 28 (1994), 285-286. [13] T.O. Banakh and R. Cauty, Topological classification of spaces of probability measures on co-analytic spaces. Math. Notes 55 (1994), 8-13. [14] T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite-Dimensional Manifolds, VNTL Publishers, Lviv (1996). [15] H.J. Baues, Homotopy types. Handbook of Algebraic Topology, I.M. James, ed., North-Holland, Amsterdam (1995), 1-72. [16] H.J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, De Gruyter, BerUn (1991). [17] M. Bell and A. Chigogidze, Topological groups homeomorphic to products of discrete spaces. Topology Appl. 53 (1993), 67-73. [18] C. Bessaga and A. Pelczynski, The estimated extension theorem, homogeneous collections and skeletons and their applications to the topological classification of linear metric spaces and convex sets. Fund. Math. 69 (1970), 153-190. [19] C. Bessaga and A. Pelczynski, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa (1975). [20] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (380) (1988). [21] M. Bestvina, PL. Bowers, J. Mogilski and J. Walsh, Characterizing Hilbert space manifolds revisited. Topology Appl. 24 (1986), 53-69. [22] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J. 33 (1986), 291-313. [23] K. Borsuk, Uber Isomorphic der Funktionalrdume, Bull. Int. Acad. Polon. Sci. et Lettres (1933), 1-10. [24] K. Borsuk, Theory of Retracts, PWN, Warszawa (1967). [25] K. Borsuk, Theory of Shape, PWN, Warszawa (1975). [26] K. Borsuk and S. Ulam, On symmetric products of topological spaces. Bull. Amer. Math. Soc. 37 (1931), 875-882. [27] H.G. Bothe, Fine Finbettung m-dimensionalen Mengen in einen (m -\- \)-dimensionalen absoluten Retrakt, Fund. Math. 52 (1963), 209-224. [28] W. Browder, J. Levine and G.R. Livesay, Finding a boundary for an open manifold, Amer. J. Math. 87 (1965), 1017-1028. [29] R. Cauty, Une caracterisation des retractes absolus de voisinage. Fund. Math. 144 (1994), 11-22. [30] R. Cauty, Un espace metrique line aire qui n'est las un retracte absolu. Fund. Math. 146 (1994), 85-99. [31] R. Cauty, T. Dobrowolski, H. Gladdines and J. van Mill, Les hyperspaces des retractes absolus et des retractes absolus de voisinage duplan. Fund. Math. 148 (1995), 257-282. [32] T.A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds. Trans. Amer. Math. Soc. 154 (1971), 399^26. [33] T.A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape. Fund. Math. 76(1972), 181-193. [34] T.A. Chapman, On the structure of Hilbert cube manifolds, Compositio Math. 24 (1972), 329-353.

368

A. Chigogidze

[35] T. A. Chapman, Cell-like mappings of Hilbert cube manifolds: Solution of a handle problem. Topology Appl. 5 (1975), 123-145. [36] T.A. Chapman, Lectures on Hilbert Cube Manifolds, Regional Series in Math., Vol. 28, Amer. Math. Soc, Providence, RI (1976). [37] T.A. Chapman, Simple homotopy theory ofANR's, Topology Appl. 7 (1977), 165-174. [38] T.A. Chapman and S. Ferry, Obstructions tofiniteness in the proper category. Unpublished manuscript. [39] T.A. Chapman and L.C. Siebenmann, Finding a boundary for a Hilbert cube manifolds. Acta Mathematica 137(1976), 171-208. [40] A. Chigogidze, Trivial bundles with Tychonov cube fiber. Mat. Notes 39 (1986), 747-756 (Russian). [41] A. Chigogidze, Noncompact absolute extensors in dimension n, n-soft mappings and their applications. Math. USSR Izv. 28 (1987), 151-174. [42] A. Chigogidze, On the structure of non-metrizable AE(0)-spaces, Mat. Notes 41 (1987), 406-411 (Russian). [43] A. Chigogidze, Compacta lying in the n-dimensional universal Menger compactum and having homeomorphic complements in it. Math. USSR Sb. 61 (1988), 471-484. [44] A. Chigogidze, The theory ofn-shapes, Russian Math. Surveys 44 (1989), 145-174. [45] A. Chigogidze, Trivial bundles and near homeomorphisms. Fund. Math. 132 (1989), 89-98. [46] A. Chigogidze, n-shapes and n-cohomotopy groups of compacta. Math. USSR Sb. 66 (1990), 329-342. [47] A. Chigogidze, Locally convex linear topological spaces that are homeomorphic to the powers of the real line, Proc. Amer. Math. Soc. 113 (1991), 599-609. [48] A. Chigogidze, Classification theorem for Menger manifolds, Proc. Amer. Math. Soc. 116 (1992), 825832. [49] A. Chigogidze, UV^-equivalence and n-equivalence. Topology Appl. 45 (1992), 283-291. [50] A. Chigogidze, Finding a boundary for a Menger manifold, Proc. Amer. Math. Soc. 121 (1994), 631-640. [51] A. Chigogidze, Inverse Spectra, North-Holland, Amsterdam (1996). [52] A. Chigogidze, Compact groups and absolute extensors. Topology Proc. 22 (1997), 63-70 [53] A. Chigogidze, Cohomological dimension of Tychonov spaces. Topology Appl. 79 (1997), 197-228. [54] A. Chigogidze, Topological characterization of torus groups. Topology Appl. 94 (1999), 13-25. [55] A. Chigogidze, Compactifications and universal spaces in extension theory, Proc. Amer. Math. Soc. 128 (2000), 2187-2190. [56] A. Chigogidze and V. V. Fedorchuk, On some dimensional properties of4-manifolds. Topology Appl. 107 (2000), 67-78. [57] A. Chigogidze, K. Kawamura and R.B. Sher, Finiteness results in n-shape theory. Topology Appl. 74 (1996), 3-16. [58] A. Chigogidze, K. Kawamura and E.D. Tymchatyn, Menger manifolds, Continua with the Houston Problem Book, Lecture Notes in Pure and Appl. Math., Vol. 170 (1995), 37-88. [59] A. Chigogidze, K. Kawamura and E.D. Tymchatyn, Nobeling spaces and pseudo-interiors of Menger compacta. Topology Appl. 68 (1996), 33-65. [60] A. Chigogidze and J.R. Martin, Fixed point sets of Tychonov cubes. Topology Appl. 64 (1995), 201-218. [61] A. Chigogidze and M. Zarichnyi, On absolute extensors modulo a complex. Topology Appl. 86 (1998), 169-178. [62] M.M. Cohen, A Course in Simple-Homotopy Theory, Springer, Berlin (1970). [63] D.W. Curtis, Hyperspaces homeomorphic to Hilbert space, Proc. Amer. Math. Soc. 75 (1979), 126-130. [64] D.W. Curtis and R.M. Schori, Hyperspaces ofPeano continua are Hilbert cubes. Fund. Math. 101 (1978), 19-38. [65] J.J. Dijkstra, J. van Mill and J. Mogilsld, Classification of finite-dimensional pseudo-boundaries and pseudo-interiors. Trans. Amer. Math. Soc. 332, (1992), 693-709. [66] T. Dobrowolski, W. Marciszewsld and J. Mogilski, On topological classification of function spaces Cp (X) of low Borel complexity. Trans. Amer. Math. Soc. 328 (1991), 307-324. [67] T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces. Open Problems in Topology, J. van Mill and G.M. Reed, eds, North-Holland, New York (1990), 409^29. [68] T. Dobrowolski and L.R. Rubin, The hyperspace of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic. Pacific J. Math. 164 (1994), 15-39. [69] T. Dobrowolski and L.R. Rubin, The space ofANR's in i?'^. Fund. Math. 146 (1994), 31-58.

Infinite dimensional topology and shape theory

369

[70] T. Dobrowolski and K. Sakai, Spaces of measures on metrizable spaces. Topology Appl. 72 (1996), 215258. [71] T. Dobrowolski and H. Toruiiczyk, On metric linear spaces homeomorphic to I2 and compact convex sets homeomorphic to Q, Bull. Acad. Polon. Sci. 27 (1979), 883-887. [72] A.N. Dranishnikov, Absolute extensors in dimension n and n-soft mappings raising dimension, Uspekhi Mat. Nauk 39 (1984), 55-95. [73] A.N. Dranishnikov, Universal Menger compacta and universal mappings. Mat. Sbomik 129 (1986), 121139. [74] A.N. Dranishnikov, On a problem of P. S. Alexandrov, Mat. Sbomik 135 (1988), 551-557. [75] A.N. Dranishnikov, Extension of mappings into CW-complexes, Mat. Sbomik 182 (1991), 47-56. [76] A.N. Dranishnikov, The Eilenberg-Borsuk theorem for maps in an arbitrary complex. Mat. Sbomik 185 (1994), 81-90. [77] A.N. Dranishnikov, On extension theory of compact space. Preprint (1998). [78] A.N. Dranishnikov, J. Dydak and J. Keesling, Extension of mappings. Preprint. [79] A.N. Dranishnikov and D. Repovs, Cohomological dimension with respect to perfect groups. Topology Appl. 74 (1996), 123-140. [80] A.N. Dranishnikov and E.V. Scepin, Cell-like mappings, Uspekhi Mat. Nauk 41 (1986), 49-90. [81] J. Dugundji, An extension ofTietze's theorem. Pacific J. Math. 1 (1951), 353-367. [82] J. Dydak, Cohomological dimension and metrizable spaces. II, Trans. Amer. Math. Soc. 348 (1996), 16471661. [83] J. Dydak, Cohomological dimension theory. Handbook of Geometric Topology, R.J. Davermann and R.B. Sher, eds, Elsevier, Amsterdam (2002), 423-^70. [84] J. Dydak and J. Segal, Shape Theory, Lecture Notes Math., Vol. 688 (1978). [85] J. Dydak and J. Segal, A list of open problems in shape theory. Open Problems in Topology, J. van Mill and G.M. Reed, eds, North-Holland, New York (1990), 457-467. [86] J. Dydak and J. Walsh, Spaces without cohomological dimension preserving compactifications, Proc. Amer. Math. Soc. 113 (1991), 1155-1162. [87] J. Dydak and J. Walsh, Infinite-dimensional compacta having cohomological dimension two: An application of the Sullivan conjecture. Topology 32 (1993), 93-104. [88] R.D. Edwards, Problems: 1987 Western Workshop in Geometric Topology, Oregon State Univ. Press (1987), 46-48. [89] R.D. Edwards, Characterizing infinite-dimensional manifolds topologically (after Henryk Torunczyk), Lect. Notes Math., Vol. 540 (1999), 278-302. [90] D.A. Edwards and R. Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. 101 (1975), 521-535. [91] VV Fedorchuk, Covariant functors in the category of compacta, absolute retracts and Q-manifolds, Russian Math. Surveys 36 (3) (1981), 211-233. [92] V.V Fedorchuk, On open mappings, Uspekhi Mat. Nauk 37 (1982), 187-188 (Russian). [93] S. Ferry, The homeomorphism group of a compact Hilbert cube manifold is an ANR, Ann. Math. 106 (1977), 101-119. [94] S. Ferry, UV''-equivalent compacta. Lecture Notes Math., Vol. 1283 (1987), 88-114. [95] R.H. Fox, On the Lustemik-Schnirelman category, Ann. of Math. 42 (1941), 333-370. [96] R. Geoghegan, Open problems in infinite-dimensional topology. Topology Proc. 4 (1979), 287-330. [97] R. Geoghegan and W. Haver, On the space of piecewise linear homeomorphisms of a manifold, Proc. Amer. Math. Soc. 55 (1976), 145-151. [98] R. Geoghegan and R. Summerhill, Pseudo-boundaries and pseudo-interiors in Euclidean space and topological manifolds. Trans. Amer. Math. Soc. 194 (1974), 141-165. [99] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, New York (1960). [100] O. Hanner, Some theorems on absolute neighborhood retracts, Arkiv Mat. 1 (1951), 389-408. [101] O. Hanner, Retractions and extensions of mappings of metric and non-metric spaces, Arkiv Mat. 2 (1952), 315-360. [102] W.E. Haver, Mappings between ANR's that are fine homotopy equivalences. Pacific J. Math. 58 (1975), 457-461.

370

A. Chigogidze

[103] R. Haydon, On a problem of Petczyriski: Milutin spaces, Dugundji spaces andAE(0 — dim), Studia Math. 52(1974), 23-31. 104] D.W. Henderson, Open subsets ofHilbert space, Compositio Math. 21 (1969), 312-318. 105] D.W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space. Topology 9 (1970), 25-34. 106] D.W. Henderson and R.M. Schori, Topological classification of infinite-dimensional manifolds by homotopy type. Bull. Amer. Math. Soc. 76 (1970), nX-UA. 107] M.I. Kadec, On topological equivalence of separable Banach spaces, Soviet Math. Dokl. 7 (1966), 319322. 108] J. Keesling and D. Wilson, The group of PL-homeomorphisms of a compact PL-manifold is an Ij manifold. Trans. Amer. Math. Soc. 193 (1975), 249-256. 109] O.H. Keller, Die Homeoomorphie der compakten convexen Mengen im Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. 110] V.L. Klee, Some topological properties of convex sets. Trans. Amer. Math. Soc. 78 (1955), 30-45. I l l ] V.I. Kuzminov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1-45. 112] S. Lefshetz, On compact spaces, Ann. Math. 32 (1931), 521-538. 113] R. Luke and W. Mason, The space of autohomeomorphisms on a compact two-manifold is an absolute neighborhood retract. Trans. Amer. Math. Soc. 164 (1972), 275-285. 114] S. MacLane and J.H.C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. USA 36 (1950), 41^8. 115] S. Mardesic and J. Segal, Shape Theory, North-Holland, Amsterdam (1982). 116] K. Menger, Allgemeine Raume und Cartesische Raume. Zweite Mitteilung: "Uber umfas-sendste n-dimensional Mengen", Proc. Akad. Amsterdam 29 (1926), 1125-1128. 117] J. van Mill, A rigid space X for which X x X is homogeneous; an application of infinite-dimensional topology, Proc. Amer. Math. Soc. 83 (1981), 597-600. 118] J.W Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280. 119] M. Mimura, Homotopy theory of Lie groups. Handbook of Algebraic Topology, North-Holland, Amsterdam (1995), 951-991. 120] J. Mogilski, Characterizing the topology of infinite dimensional a-compact manifolds, Proc. Amer. Math. Soc. 92 (1984), 111-118. 121] G.M. Nepomnyashchii, On the spectral decomposition of the multi-valued absolute retracts, Uspekhi Mat. Nauk 36 (1981), 221-222 (Russian). 122] W. Olszewski, Completion theorem for cohomological dimensions, Proc. Amer. Math. Soc. 123 (1995), 2261-2264. 123] W. Olszewski, Universal separable metrizable spaces of given cohomological dimension. Topology Appl. 61 (1995), 293-299. 124] L.S. Pontryagin, Topological Groups, Princeton Univ. Press, Princeton, NJ (1946). 125] T. Porter, Proper homotopy theory. Handbook of Algebraic Topology, I.M. James, ed., North-Holland, Amsterdam (1995), 127-167. 126] E.V. Scepin, Topology of limit spaces of uncountable inverse spectra, Uspekhi Mat. Nauk 31 (1976), 191226 (Russian). 127] E.V. Scepin, Finite-dimensional bicompact absolute neighborhood retracts are metrizable, Dokl. Akad. Nauk SSSR 233 (1977), 304-307. 128] E.V. Scepin, On Tychonov manifolds, Dokl. Akad. Nauk SSSR 246 (1979), 551-554 (Russian). 129] R.M. Schori, Topological stability for infinite-dimensional manifolds, Compositio Math. 23 (1971), 87100. 130] S. Shelah, Infinite Abelian groups, Whitehead problem and some consequences, Israel J. Math. 18 (1974), 243-256. 131] L.C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five. Doctoral dissertation, Princeton Univ. Press, Princeton, NJ (1965). 132] L.C. Siebenmann, L'invariance topologique du type simple d'homotopic, Seminaire Bourbaki, 25^ anee, n° 428 (1972/73).

Infinite dimensional topology and shape theory

371

[133] W. Sierpinski, Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnee, C. R. Acad. Paris 162 (1916), 629-632. [134] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). [135] M.A. Stanko, Solution ofMenger's problem in the class ofcompacta, Soviet Math. Dokl. 12 (1971), 18461849. [136] H. Torunczyk, Ph.D. Thesis, Warsaw. [137] H. Torunczyk, Absolute retracts as factors ofnormed linear spaces, Fund. Math. 86 (1974), 53-67. [138] H. Torunczyk, Homeomorphism groups of compact Hilbert cube manifolds that are manifolds. Bull. Pol. Acad. Sci. 25 (1977), 401^08. [139] H. Torunczyk, Concerning locally homotopy negligible sets and characterization of l2-manifolds. Fund. Math. 101 (1978), 93-110. [140] H. Torunczyk, On CE-images of the Hilbert cube and characterization of Q-manifolds. Fund. Math. 106 (1980), 3 1 ^ 0 . [141] H. Torunczyk, Characterizing Hilbert space topology. Fund. Math. I l l (1981), 247-262. [142] H. Torunczyk and J.E. West, Fibrations and bundles with Hilbert cube manifold fibers, Mem. Amer. Math. Soc. 80 (406) (1989). [143] V.S. Varadarajan, Measures on topological spaces. Mat. Sbomik 55 (1961), 35-100. [144] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56-69. [145] J.J. Walsh, Dimension, cohomological dimension and cell-like mappings. Lecture Notes Math., Vol. 870 (1981), 105-118. [146] J.E. West, Infinite products which are Hilbert cubes. Trans. Amer. Math. Soc. 150 (1970), 1-25. [147] J.E. West, The ambient homeomorphy of an incomplete subspace of infinite-dimensional Hilbert spaces. Pacific J. Math. 34 (1970), 257-267. [148] J.E. West, Mapping Hilbert cube manifolds to ANR's. A solution of a conjecture of Borsuk, Ann. Math. 106 (1977), 1-18. [149] J.E. West, Open problems in infinite dimensional topology. Open Problems in Topology, J. van Mill and G.M. Reed, eds, North-Holland, New York (1990), 523-597. [150] J.H.C. Whitehead, On the homotopy type of ANR's, Bull. Amer. Math. Soc. 54 (1948), 1133-1145. [151] J.H.C. Whitehead, Combinatorial homotopy I, Bull. Amer. Math. Soc. 55 (1949), 213-245. [152] G.T. Whybum, Topological characterization of Sierpinski curve. Fund. Math. 45 (1958), 320-324. [153] M. Wojdyslawski, Retractes absolus et hyperspaces des continus. Fund. Math. 32 (1939), 184-192.

This Page Intentionally Left Blank

CHAPTER 8

Nonpositive Curvature and Reflection Groups Michael W. Davis* Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210, USA E-mail: [email protected]

Contents Introduction 1. Nonpositively curved spaces 1.1. The CAr(e)-inequality 1.2. Piecewise constant curvature polyhedra 1.3. The CAT(l) condition for links 1.4. Infinitesimal shadows and the ideal boundary 1.5. Some well-known results in geometric topology and their implications for CAr(O) spaces 1.6. Euler characteristics and the Combinatorial Gauss-Bonnet Theorem 2. Coxeter groups 2.1. Coxeter systems 2.2. Coxeter cells 2.3. The cell complex U (in the case where W is infinite) 2.4. Reflection groups 2.5. Applications and examples 3. Artin groups 3.1. Hyperplane complements 3.2. The Salvetti complex 3.3. Complexes of groups 3.4. Reinterpretation of the Main Conjecture 3.5. A cubical structure on

*Partially supported by NSF Grants DMS-9208071 and DMS-9505003. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved 373

375 375 376 378 380 386 388 392 395 395 397 398 400 401 408 408 411 414 417 419 420

This Page Intentionally Left Blank

Nonpositive curvature and reflection groups

375

Introduction A space is aspherical if its universal cover is contractible. Examples of aspherical spaces occur in differential geometry (as complete Riemannian manifolds of nonpositive sectional curvature), in Lie groups (as r\G/K where G is a Lie group, A' is a maximal compact subgroup and T is a discrete torsion-free subgroup), in 3-manifold theory and as certain 2-dimensional cell complexes. The main purpose of this paper is to describe another interesting class of examples coming from the theory of reflection groups (also called "Coxeter groups"). One of the main results is explained in Sections 1.3 and 2.3: given a finite simplicial complex L, there is a compact aspherical polyhedron X such that the link of each vertex in X is isomorphic to the barycentric subdivision of L. A version of this result first appeared in [29]. Later, in [45], Gromov showed that the polyhedron X can be given a piecewise Euclidean metric which is nonpositively curved in the sense of Aleksandrov. This gives a new proof of the asphericity of X (cf. Theorem 1.5 below). As we vary the choice of the link L we get polyhedra X with a variety of interesting properties. For example, if L is homeomorphic to an (n — 1)-sphere, then X is an /i-manifold. More examples are discussed in Section 2.5. Section 1 covers background material on nonpositive curvature. The main examples are discussed in Section 2. Section 3 deals with some related aspherical complexes which arise in the study of complements of arrangements of hyperplanes. This paper began as a set of notes for three lectures which I gave at the Eleventh Annual Workshop in Geometric Topology at Park City, Utah in June 1994. In the course of preparing it for this volume I have added approximately 25 percent more material, notably. Sections 1.4, 1.5, 2.4 and parts of Section 2.5.1 would like to thank Lonette Stoddard for preparing the figures.

1. Nonpositively curved spaces The notion of "nonpositive curvature" (or more generally of "curvature bounded above by a real number f") makes sense for a more general class of metric spaces than Riemannian manifolds: one need only assume that any two points can be connected by a geodesic segment. For such spaces, the concept of curvature bounded above by e can be defined via comparison triangles and the so-called "CAr(6:)-inequality". (This terminology is due to Gromov.) This is explained in Sectionl.l. In Section 1.2 we consider "piecewise constant curvature polyhedra" and give a condition (in terms of hnks of vertices) for such a polyhedron to have curvature bounded from above. The condition is that each link be CAT (I). In Section 1.3 we discuss criteria for such a link to be CAT (I). The two conditions we are most interested in are given in Gromov's Lemma and Moussong's Lemma. These give criteria for piecewise spherical simplicial complexes (with sufficiently big simpUces) to be CAT (I). In Section 1.4 we show that a proper CAT(0) space can be compactified by adding an ideal boundary consisting of "endpoints" of geodesic rays. In this section we also discuss the notion of an "infinitesimal shadow" which measures the nonuniqueness of geodesic continuation at a point. In Section 1.5 we sketch a proof of Theorem 1.47 which asserts that the compactification of a CAT(0) PL manifold is homeomorphic to a disk. It

376

M.W.Davis

is also indicated how such a result might fail in the non-PL context. (Explicit examples of this failure are given in Section 2.5(g).) In Section 1.6 we discuss a conjecture of H. Hopf concerning the Euler characteristic of a closed, nonpositively curved, even-dimensional manifold. Using the combinatorial version of the Gauss-Bonnet Theorem this leads us to a conjecture concerning a number associated to a piecewise spherical structure on an odd-dimensional sphere. 1.1. The CAT(e)-inequality Given a smooth Riemannian manifold M one defines its "curvature tensor" and from this its "sectional curvature". The sectional curvature A' of M is a real-valued function on the set of all pairs (jc, P) where x is a point in M and P is a tangent 2-plane at x. Given a real number e, we say that "M has curvature ^ e" and write K{M) ^ e if the sectional curvature K is bounded above by s. It has long been recognized that the condition that the curvature of M is bounded above is equivalent to a condition which can be phrased purely in terms of the underlying metric (i.e., in terms of the distance function) on M. In fact, there are several possible versions of such a condition. We shall focus on one called the ''CAT^s) condition" by Gromov. ("C" stands for either "Cartan" or "comparison", "A" of "Aleksandrov", and "T" for "Toponogov".) Once one has such a condition one can define the notion of "curvature ^ £" for many "singular" metric spaces, that is, for a more general class of metric spaces than Riemannian manifolds. Good references for this material are [44] and [15]. We begin by stating the following Comparison Theorem of Aleksandrov. A proof can be found in the article of M. Troyanov in [44]. 1.1 (Aleksandrov). Let M be a simply connected, complete Riemannian manifold and 8 a real number. Then K{M) ^ e if and only if each geodesic triangle in M {of perimeter ^ In/^) satisfies the CAT{e)-inequality. THEOREM

A "geodesic triangle" in M means a configuration in M consisting of three points (the "vertices") and three (minimal) geodesic segments connecting them (the "edges"). The term "'CATiey is explained below. As e > 0, = 0, or < 0, let M^ stand for S^ (the 2-sphere of constant curvature e) ¥? (the Euclidean plane) or H^ (the hyperbolic plane of curvature s). Let r be a geodesic triangle in M. A comparison triangle for T is a geodesic triangle T* in Ml with the same edge lengths as T. Choose a vertex x of 7 and a point y on the opposite edge. Let JC* and j * denote the corresponding points in T*. (See Figure 1.) The CAr(e)-inequality is J(jc,y)^J*(jc*,/) where d and J* denote distance in M and M^, respectively. REMARK. Sl is the sphere of radius 1/v^. Since any geodesic triangle T* c 5^, must lie in some hemisphere, we see that the perimeter of T* can be no larger than litj^ (the

Nonpositive curvature and reflection groups

?>11

Fig. 1.

circumference of the equator). So, when ^ > 0, the CAr(£)-inequahty only makes sense for triangles of perimeter ^ In/y/e. Now let (X, d) be a metric space. A path of: [A, Z?] -> Z is a geodesic if it is an isometric embedding, i.e., '\fd{a{t), oi{s)) = |/ — 5*1 for all s, t in [a, b]. DEFINITION 1.2. A metric space Z is a geodesic space (or a "length space") if an two points can be connected by a geodesic segment.

We shall also assume that X is complete and locally compact. (The hypothesis of local compactness could be removed, cf. [15].) The notion of a geodesic triangle clearly makes sense in a geodesic space as does the CAr(£)-inequality. 1.3. A geodesic space X is CAT{e) if the CAr(£)-inequality holds for all geodesic triangles T of perimeter < In/^/e and for all choices of vertex x and point y on the opposite edge. (The condition that the perimeter be ^ In/^ is interpreted to be vacuous if s ^ 0.) X has curvature ^ e, written K{X) ^ e, if it satisfies CAT{8) locally. DEHNITION

(i) If e' < e, then CAT(s') implies CAT(£) and K(X) < e' implies K(X) < s. (ii) There is a completely analogous definition of curvature bounded from below: one simply reverses the CAr(e)-inequality. (See [1] or [52], Chapter 16 in this Handbook.) REMARKS,

Some consequences of CAT(8). (i) There are no digons in X of perimeter < ITZJ^. (A digon is a configuration consisting of two distinct geodesic segments between points x and y.) The reason is that we could introduce a third vertex in the interior of one segment and obtain a triangle for which the CAr(e)-inequahty clearly fails. As special cases of this principle, we have the following. (a) If X is CAr(l), then a geodesic between two points of distance < TT is unique. (b) If X is CAr( 1), then there is no closed geodesic of length < 27r. (A closed geodesic is an isometric embedding of a circle.) (c) If X is CAr(O), then any two points are connected by a unique geodesic. (ii) IfX is CAr(O), then the distance function J : Z x Z -> [0, oo) is convex. (In general, a function (^: 7 -^ R on a geodesic space Y is convex, if given any geodesic path

378

M.W.Davis

Of: [a, Z?] ^- y the function (^ o a : [a, Z?] -> R is a convex function. In particular, Z x X, with the product metric, is a geodesic space and the statement that ^ : X x X ^- [0, oo) is convex means that given geodesic paths a\{a,b^^^ X and )S: [c, J] -> X the function {s, t) -^ d(a(s), Pit)) is a convex function on [a, b] x [c, d].) There is the following generalization of the Cartan-Hadamard Theorem for nonpositively curved manifolds. 1.4. If X is a geodesic space with convex distance function (e.g., if X is CAT(0)), then X is contractible.

PROPOSITION

PROOF. The convexity of the distance function implies that X has no digons. Hence, any two points of X are connected by a unique geodesic. Choose a base point XQ and define the contraction H:X yi I -> Xhy contracting along the geodesic to JCQ. The proof that H is continuous follows easily from the convexity of J. D

Suppose ^ ( X ) ^ s. Then since CAT{£) holds locally, X is locally convex (i.e., in any sufficiently small open set, any two points are connected by a unique geodesic). Therefore, X is locally contractible. In particular, any such X has a universal cover. REMARK.

1.5. Let e ^ 0. If X is a geodesic space with K(X) ^ e, then its universal cover X is CAT(6). (Inparticular, X is contractible.) THEOREM

This theorem is stated by Gromov in [45, p. 119] and proved in W. Ballman's article in [44, p. 193]. (Quite possibly, it was known to Aleksandrov.) REMARK. Theorem 1.5 is not true for e > 0. There is an analogous result for ^ > 0: the hypothesis of simple connectivity is unimportant, but one needs to rule out the possibility of closed geodesies of length < lixj^. A version of this is stated as Lemma 1.13, below. COROLLARY

1.6. IfK(X)

^ 0, then X is a K{7t, \)-space {i.e., X is aspherical).

1.2. Piecewise constant curvature polyhedra Let M^ stand for S^, E" or W^ as e is greater than, equal to, or less than 0, respectively. A "half-space" in 5" is a hemisphere; a "half-space" in E" or H" has its usual meaning. DEFINITION 1.7. A (convex) cell in M^ is a compact intersection of a finite number of half-spaces. (When £ > 0, one can also require that the cell does not contain a pair of antipodal points.) DEFINITION 1.8. An M^ cell complex X is a cell complex formed by gluing together cells in M^ via isometrics of their faces, {s is fixed, n can vary.) If ^ = 0, X is cdWtd piecewise Euclidean (abbreviated PE). If e = 1, X i^ piecewise spherical (abbreviated PS). If £ = — 1, X i^ piecewise hyperbolic (abbreviated PH).

Nonpositive curvature and reflection groups EXAMPLE

379

1.9. The surface of a cube is a PE complex.

If X is an M^ cell complex, then we can measure the length € of a path in X\ the length of the portion of the path within a given cell is defined using arc length in M^. The intrinsic metric £ on Z is defined as follows: d{x, y) = mf[l(w) | a is a path from jc to j } . (If X is not path connected, the d may take oo as a value.) Does the intrinsic metric give X the structure of a geodesic space? The issue is whether the infimum occurring in the definition of d can actually be realized by a minimal path. If X is locally finite and if there is a 5 > 0 so that all closed (5-balls in X are compact (e.g., if X is a finite complex), then the Arzela-Ascoli Theorem implies that X is a complete geodesic space. Links. Suppose that a is an n-cell in M^ and that i; is a vertex of a. The Riemannian metric on M^ gives an inner product on its tangent space Ty(M^) at v. The set of inward pointing directions at v is subset of the unit sphere in Ty{M^). In fact, this subset is a spherical cell, which we denote by Lk(v, a). We think of it as a cell in 5^^"^ well-defined up to isometry. (See Figure 2.) If X is an M^-complex and f is a vertex of X, then the link ofv in X is defined by

Lk(v,X)= y

Lk(v,a).

vCa

This is a PS cell complex. Thus, the link of a vertex in any Mg cell complex has a natural piecewise spherical structure. EXAMPLE 1.10. As in Example 1.9, let X be the surface of a cube and u, a vertex. Then the link of v in each square is a circular arc of length 7r/2; hence, the link of i; in X is a circle of length 37r/2.

In [45, p. 120] Gromov gave the following "infinitesimal" condition for deciding if a piecewise constant curvature polyhedron has curvature bounded from above.

Lk(v,a) Fig. 2.

380

M.W.Davis

1.11 (Aleksandrov, Gromov, Ballman). Suppose X is an Ms-cell complex. Then K(X) < e if and only if for each vertex v, Lk{v, X) is CAT(l). THEOREM

A proof of this can be found in Ballman's article in [44, p. 197]. The result must have also been known to Aleksandrov's school, since they knew that an "M^ cone" on a CAT(l) space was CAT{£). EXAMPLE 1.12. A PS structure on a circle S is CATil) if and only if l(S) ^ In. Therefore, a PE structure on a surface has A' < 0 if and only if at each vertex the sum of the angles is > 27r. For example, the surface of a cube does not have nonpositive curvature.

1.3. The CAT (I) condition for links In order to use Theorem 1.11 we need to be able to decide when the link of a vertex is CAT (I). So, suppose L is some PS cell complex. We need to be able to answer the following. QUESTION. HOW

do you tell if L is CAT(l)l

The following lemma gives an inductive procedure for studying this question. 1.13. APS complex L is CAT(l) if and only if (i) K(L)^l, and (ii) every closed geodesic in L has length ^ In.

LEMMA

By Theorem 1.11, condition (i) can be checked by looking at links of vertices in L. Thus, (ii) is the crucial condition. We next would like to explain several situations in which we have a satisfactory answer to our question. These will be grouped under the following headings: (a) Gromov's Lemma, (b) Moussong's Lemma, (c) Orthogonal joins, (d) Spherical buildings, (e) Polar duals of hyperbolic cells, (f) Branched covers of round spheres. In these notes we will mostly be concerned with the first two headings (and we will confine ourselves to a few brief comments about the other four). (a) Gromov's Lemma. Let D" denote a regular n-cubt in E'^ and let f be a vertex of D". Then Lk(v, D") is the regular spherical {n — l)-simplex A^~^ spanned by the standard basis e\,. ..,en of R". (So, Z\"~^ is the intersection of 5""^ with the positive "quadrant" [0,oo)"inR".) A spherical (n — l)-simplex isometric to Z\"~^ will be called an all right simplex. An all right simplex is characterized by the fact that all its edge lengths are n/2. Alternatively, it can be characterized by the fact that all its dihedral angles are 7T/2.

Nonpositive curvature and reflection groups DEFINITION

381

1.14. A PS simplicial cell complex is all right if each of its simplices is all

right. 1.15. If X is a PE cubical complex, then each of its links is an all right simplicial cell complex.

EXAMPLE

1.16. A simplicial complex K is 2iflag complex if any finite set of vertices, which are pairwise joined by edges, span a simplex in K. DEFINITION

Combinatorialists use "cHque complex" instead of "flag complex". Alternative terminology, which has been used elsewhere, is that K is "determined by its 1-skeleton", or K has "no empty simplices", or K satisfies the "no Z\-condition". (The last is Gromov's terminology.) The term "flag complex" is taken from [17]. REMARK 1.17. Let V be a set with a symmetric and reflexive relation (an "incidence relation"). Let K be the abstract simplicial complex whose simplices are the finite subsets of V which are pairwise related. Then ^ is a flag complex. Conversely, given a flag complex K, one defines a relation on its vertex set V by saying that two vertices are related if they are joined by an edge. This relation gives back K as its associated complex.

1.18. Let P be a poset. Then if we make the order relation symmetric and take the associated simplicial complex, we get a flag complex. Its poset of simplices is denoted by V\ and called the derived complex of V. The elements of V' are finite chains (i;o < • • • < Vk) in V. EXAMPLE

EXAMPLE 1.19. If P is the poset of cells in a cell complex, then V' can be identified with the poset of simplices in its barycentric subdivision. Thus, the barycentric subdivision of any cell complex is a flag complex.

1.20. If K is the boundary of an m-gon (i.e., ^ is a circle with m edges) then A' is a flag complex if and only if m > 3.

EXAMPLE

1.21 (Gromov's Lemma). Let L be an all right, PS simplicial complex. Then L is CAT (I) if and only if it is a flag complex.

LEMMA

COROLLARY

1.22 (Berestovskii [8]). Any polyhedron has a PS structure which is

CATil). PROOF. Let L be a cell complex. By taking the barycentric subdivision we may assume that L is a flag complex. Then give L a piecewise spherical structure by declaring each simplex to be all right. D COROLLARY 1.23. Let X be a PE cubical complex. Then K(X) ^ 0 if and only if the link of each vertex is a flag complex.

382

M.W.Davis

APPLICATION 1.24 (HyperboHzation). In [45] Gromov described several functorial procedures for converting a cell complex J (usually a simplicial or cubical complex) into a PE cubical complex H(J) with nonpositive curvature. (See also [35] and [21].) H(J) is called a "hyperbolization" of J. Since H(J) is aspherical it cannot, in general, be homeomorphic to J. However, there is a natural surjection H(J) -> 7. Also, H(J) should have the same local structure as J in the following sense: the link of each "hyperbolized cell" is PL homeomorphic to the link of the corresponding cell in J. Usually, the new link will be the barycentric subdivision of the old one (or else the suspension of a barycentric subdivision of an old Unk). Thus, the new links will be flag complexes and Gromov's Lemma can be used to prove that H{J) is nonpositively curved. (A different argument is given in [35] and [45].)

The proof of Gromov's Lemma is based on the following. SUBLEMMA 1.25 ([45, p. 122]). Let v be a vertex in an all right, PS simplicial complex and let B be the closed ball of radius njl about v (i.e., B is the closed star ofv). Let x, y be points in dB (the sphere of radius 7t/2 about v) and let y be a geodesic segment from x to y such that y intersects the interior of B. Then t(y) ^n. Let A be an all right simplex in B with one vertex at v and suppose that y intersects the interior of A. Consider the union of all geodesic segment which start at v, pass through a point in y fl Z\ and end on the face of A opposite to u. It is an isoceles spherical 2-simplex with two edges of length TT/I as in Figure 3. (Think of a spherical 2-simplex with one vertex at the north pole and the other two on the equator.) Let Q be the union of all these 2-simplices. Then Q can be "developed" onto the northern hemisphere of S^. If t(y) < 7r, then Q is isometric to a region of S^ so that i; maps to the north pole and x and y to points on the equator. But if two points on the equator of S^ are of distance < n, then the geodesic between them is a segment of the equator. This contradicts the hypothesis that the image of y intersects the open northern hemisphere. D PROOF.

Let L be an all right, PS simplicial cell complex. First suppose that L is not a flag complex. Then either L or the link of some simplex of L PROOF OF GROMOV'S LEMMA.

spherical 2-simplex

Fig. 3.

Nonpositive curvature and reflection groups contains an "empty" triangle. Such a triangle is a closed geodesic of length 37r/2 (which is < In). Hence, L is not CAT {I). Conversely, suppose that L is a flag complex. Then the link of each vertex is also a flag complex and by induction on dimension we may assume that K{L) ^ 1. Hence, it suffices to show every closed geodesic in L has length ^ In. Suppose, to the contrary, that a is a closed geodesic with t{a) < lit. Let L' be the full subcomplex of L spanned by the set of vertices v such that a fi Star(i;) ^ 0. (Here Star(i;) denotes the open star of v.) By Sublemma 1.25, a cannot intersect two disjoint open stars. Hence any two vertices of L' must be connected by an edge. Since L is a flag complex, this implies L' is an all right simplex. But this is impossible since a simplex contains no closed geodesic. D (b) Moussong 's Lemma DEFINITION

1.26. A spherical simplex has size >

TT/I

if all of its edge lengths are > n/2.

1.27. Let L be a PS simplicial complex with simplices of size ^ n/l. L is a metric flag complex if given a set of vertices {i;o, •.., i^jt}, which are pairwise joined by edges, such that there exists a spherical /:-simplex with these edge lengths, then [VQ, ...,Vk} spans a /:-simplex in L. DEFINITION

LEMMA 1.28 (Moussong's Lemma). Let L be a PS simplicial complex with simplices of size ^ 7r/2. Then L is CAT{\) if and only if it is a metric flag complex.

This generalization of Gromov's Lemma is the main technical result in the Ph.D. thesis of Moussong [51]. Its proof is quite a bit more difficult than that of Gromov's Lemma and we will not try to explain it here. We will use it in the next chapter to show that a certain PE complex associated to any Coxeter group is CAr(O). DEFINITION

1.29. A cell is simple if the link of each vertex is a simplex.

The edge lengths of such a simplex are interior angles in the 2-dimensional faces. Thus, such a simplex has size > n/l if all such angles in the 2-cells are ^ 7r/2. COROLLARY 1.30. Let X be a PE complex with simple cells and with 2-cells having nonacute angles. Then K {X) ^ 0 if and only if the link of each vertex is a metric flag complex.

(c) Orthogonal joins. Suppose that ai c 5^^ and ^2 C S^^ are spherical cells. Regard 5^' and 5^2 as a pair of orthogonal great subspheres in 5^1+^2+1 (^ 3^^1+1 x M^2+i Then the orthogonal join o\ * 02 of o\ and a^ is the union of all geodesic segments from o\ to 02 in ^/:i+/:2+i jt is naturally a spherical cell of dimension equal to dimai + dimor2 + 1. If Li and L2 are PS cell complexes, then their orthogonal join Li * L2 is defined to be the union of all cells a\ * 02 where a\ is a cell in L\ and 02 is a cell in L2. It is naturally a PS cell complex, homeomorphic to the usual topological join of the underlying polyhedra. The following result is proved in the appendix of [20].

383

384

MM Davis

PROPOSITION

1.31. IfL\ andL2 are CAT(l), PS cell complexes, then L\ *L2 is CAT{\).

1.32. For example, taking L2 to be a point we see if Li is CAT{\), then so is the "spherical cone" on Li. Similarly, taking L2 = 5^, we see that the "spherical suspension" ofLi isCAr(l).

REMARK

(d) Spherical buildings. Tits has defined a certain remarkable class of simplicial complexes called "buildings", e.g., see [17] and [55]. Associated to a building B there is a Coxeter group W. (This will be defined in Section 2.) The building B can be written as a union of apartments AQ^ ,

5 = U^«, where each Aa is isomorphic to the Coxeter complex for W. If W is a finite group, then this Coxeter complex can naturally be thought of as a triangulation of 5"", the round n-sphere, for some n. The building is called spherical if its associated Coxeter group is finite (so that each apartment is a round sphere). Thus, a spherical building has a natural structure of a PS simplicial complex. The axioms for buildings imply that any two points in B lie in a common apartment. Furthermore, the geodesic between them also lies in this apartment. From this we can immediately deduce the following. (See also [32].) THEOREM

1.33. Any spherical building is CAT {I).

EXAMPLE 1.34. A generalized m-gon is a connected, bipartite graph of diameter m and girth 2m. (A graph is bipartite if its vertices can be partitioned into two sets so that no two vertices in different sets span an edge. The diameter of a graph is the maximum distance between two vertices, its girth is the minimum length of a circuit.) A 1-dimensional spherical building is the same thing as a generalized m-gon (m ^ 00). The piecewise spherical structure is defined by declaring each edge to have length n/m.

(e) Polar duals of hyperbolic cells. Suppose that C" is a convex n-cell in hyperbolic n-space W. Let F be a face of codimension ^ in C", k ^ I. Choose a point x in the relative interior of F and consider the unit sphere S^~^ in the tangent space r^IHI". The set of outward-pointing unit normals to the codimension one faces which contain F span a spherical (k — l)-cell in 5"~^ which we denote by cr/7. (Roughly, cff is the set of all outward pointing unit normals at F.) The polar dual of C" is defined as

P(C«) = U a . . It is a PS cell complex, which, it is not difficult to see, is homeomorphic to 5"" K For further details, see [23].

Nonpositive curvature and reflection groups

385

REMARK 1.35. (i) The same construction can be carried out in E'^ or S^. For a cell in E" its polar dual is a PS cell complex which is isometric to the round (n — 1)-sphere. For a cell C" in S^, its polar dual is just the boundary of the dual cell C*, where C* = {x e S^ \ d(x,C^) ^ 7T/2}. In all three cases, the cell structure on P(C^) is combinatorially equivalent to the boundary to the dual polytope to C^. (ii) If we use the quadratic form model for W, and C" c H", then PiC^) is naturally a subset of the unit pseudosphere, S^ = {x e W'^^ \ {x, x) = I], where (x,x) = —(x\)^ H-

THEOREM

1.36. Suppose C is a convex cell in M". Then P(C) is CAT(l),

When n = 2, P{C^) is a circle, the length of which is the sum of the exterior angles of C^. By the Gauss-Bonnet Theorem, this sum is lit -h Area(C^). This completes the proof for n = 2. When « = 3, the theorem is due to Rivin and Hodgson [53]. For n > 3, it appears in [23]. FURTHER REMARKS 1.37. (i) A stronger result is actually true. The length of any closed geodesic in P{C^) is strictly greater than 2n. (As we saw, for n = 2, this follows from the Gauss-Bonnet Theorem.) Furthermore, the same is true for the link of every cell in P(C") (since such a link is, in fact, the polar dual of some face of C"). Sometimes I have defined Si PS cell complex to be "large" if it is CAT (I). Perhaps PS complexes satisfying the above stronger condition should be "extra large". (ii) The definition of polar dual makes sense for any intersection of half-spaces in H" (compact or not) and it is proved in [23] that these polar duals are also CAT (I). (iii) The main argument of [53] is in the converse direction. They show that any PS structure on S^ which is extra large arises as the polar dual of a 3-cell in H^ (unique up to isometry). An analogous result, relating metrics with K ^ I on S^ to convex surfaces in E^ had been proved much earlier by Aleksandrov. (iv) My main interest in Theorem 1.36 is that it provides a method for constructing a large number of examples of CAT (I), PS structures on 5""^, which are not covered by Moussong's Lemma. Moreover, if we deform a convex cell in H" we obtain a large family of deformations of its polar dual through CAT (I) structures. (v) A nice application of Theorem 1.36 is given in [26]. Identify M.^ with the sheet of the hyperboloid in R""^^ defined by (x, x) = — 1 and jc„+i > 0. Let V be any discrete subset of H" such that each Dirichlet domain for V is bounded. Take the convex hull of V in R'^"^* and let B(V) denote its boundary. It is not hard to see that the restriction of the bilinear form to the tangent space of any face in B(V) is positive definite and hence, that B(V) has a natural PE structure. Moreover, for any vertex v in V, the link of v in B(V) can be identified with the polar dual of the Dirichlet cell centered at v. So, by Theorem 1.36, B(V) is CAT(0). As a corollary we have that any complete hyperboHc manifold has a nonpositively curved, PE structure.

(f) Branched covers of round spheres. Suppose that M" is a smooth Riemannian manifold and that p:M^ ^^ M^ is a branched covering by some other manifold M^. The metric on M" induces a (non-Riemannian) metric on M '^.

386

MM Davis

QUESTION.

If M" has sectional curvature ^ e, then when is K(M^) ^ el

We further suppose that the branching is locally modeled on M" -^ R"/G where G is some finite linear group. (Since M" is a manifold we must therefore have that W/G is homeomorphic to R".) The following two conditions are easily seen to be necessary for K(M) ^ e:

(i)

K(M)^8,

(ii) locally, the closure of each stratum of the branched set is a convex subset of M. Let jc G M be a branch point and let Sx be the unit sphere in T^M. There is an induced finite sheeted branched cover Sx -^ Sx. Since the branched set in Sx must satisfy (ii), it follows that the metric on Sx (induced from the round metric on Sx) is piecewise spherical. We think to Sx as the "link" at a point x e p~^(x). It turns out ([20, Theorem 5.3]) that together with (i) and (ii) the following condition is necessary and sufficient for K (M) to be ^ s: (iii) Sx is CAT (I), for all branch points JC. Therefore, the answer to our question is closely tied to the question of when the branched cover of a round sphere is CAT (I). A detailed study of this question is made in [20]. For example, suppose that G is a finite, noncyclic subgroup of S0(3) (so that G is either dihedral or the group of orientation-preserving symmetries of a regular solid). Then S^/G is homeomorphic to S-^ and 5^ -^ S^/G has three branch points. Choose three points jci, X2 and JC3 in the round 2-sphere S^ and assign x/ a branching order of m,, where ^ ( 1 / m / ) > 1. Let S'^ -^ 5^ be the corresponding |G|-fold branched cover. In [20] we prove the following result. 1.38. J^ is CAT{\) if and only if (i) x\,X2, and X3 lie on a great circle in 5^, but are not contained in any semi-circle, and (ii) d{xi,Xj) ^ 71 /mk, where (/, j , k) is any permutation 6>/(l, 2, 3).

PROPOSITION

1.4. Infinitesimal shadows and the ideal boundary Suppose that Z is a piecewise constant curvature polyhedron and that c: {—a, a) -> X is a piecewise geodesic segment. Subdividing, we get two piecewise geodesic arcs <^out • [0, a) ^^ X and cin: [0, a) -^ X, defined by Cout(0 = ^(0 and Cin(0 = c(—t), and two unit tangent vectors c^^ and c[^ in Lk(c(0), X). The proof of the following lemma is left as a straightforward exercise for the reader. 1.39. A piecewise geodesic path c: (—a, a)-^ X is a local geodesic at c(0) (i.e., its restriction to some smaller interval about 0 is a geodesic) if and only if the distance from c[^ to CQUJ in Lk{c(0), X) is ^n. LEMMA

EXAMPLE 1.40. Let 5^ (In + 8) denote a circle of circumference 27t-{-8, and let X denote the Euclidean cone on S^ {In + s). Thus, x = ([0, 00) x S^ (In + 8))/ '-, where (r, 0) -(r^ 0^) if and only if r = r' = 0. The metric is given by the usual formula for metric on R^

Nonpositive curvature and reflection groups

387

in polar coordinates (see [15]). Let 0i,02 e S^(lit + 8) be two points with d(0\,02) ^ TT and consider the path c: R -> X defined by

^^~\it,02)

forf ^ 0 .

It follows from 1.39, that c is a geodesic in X. Thus, if 5 > 0, there are many possible ways to continue the geodesic ray c|(_oo,o] to a geodesic line. These continuations are parametrized by an arc of ^2's, in fact, by the arc of radius ^8 about the point of distance 7t + ^8 from^i. This example illustrates a dramatic difference between singular metric spaces (e.g., piecewise constant curvature polyhedra) and Riemannian manifolds: in the singular case extensions of geodesic segments are not necessarily unique. Suppose that X is a piecewise constant curvature polyhedron, that x e X and that V € Lk(v, X). The infinitesimal shadow of x with respect to v, denoted by Shad(x, v), is the subset of Lk(x, X) consisting of all w e Lk(x, X) such that there is a geodesic c: (—a, a)^^ X with c(0) =x,c[^ = v and CQ^^ = w. Thus, Shad(jc, v) measures the possible outgoing directions of possible extensions for a geodesic coming into x from the direction v. In particular, X has "extendible geodesies" if and only if Shad(jc, v) is nonempty for all choices of x and v. For example, if Lk(x, X) is isometric to the round sphere S^~^, then it follows from Lemma 1.39 that the infinitesimal shadow of jc with respect to v is the antipodal point —v. As another example, suppose that X is the cone on S^(2n -\- 8), 8 ^ 0, as in Example 1.40. Then any infinitesimal shadow at the cone point is an arc of radius ^8 in S^ {In -h 8). Combining this example with Lemma 1.39, we get the following result. LEMMA 1.41. Suppose that X is a piecewise constant curvature polyhedron, that x eX and that v e Lk(x, X). Then

Shad(x, v)=Lk(x,

X) - Bjr(v),

where Bj^ {v) denotes the open ball in Lk(x, X) of radius n about v. The ideal boundary. Now suppose that X is a proper CAT{0) space. ("Proper" means that closed metric balls are compact.) Then X can be compactified to a space X by adding an "ideal boundary" X(oo). Here is the idea. Fix a base point JCQ G X. We compactify X by adding an endpoint c(oo) to each geodesic ray c: [0, oo) -> X, with c(0) = XQ. Thus, X(oo) is the set of geodesic rays beginning at the base point. The topology on X can be described as follows. Let z = c(oo) e X(oo) and let U be an open neighborhood of c(r) in SXQ (r) (the sphere of radius r about XQ) and let V be the set of all points of the form b(t), t e (r, oo], where Z?: [0, oc) -^ X is a geodesic ray starting at xo and passing through U. The sets V a neighborhood basis at z. At a point JC G X, a neighborhood basis consists of the open balls centered at x.

388

M.W.Davis

For J* > r, by using geodesic contraction, one can define a natural projection

from one closed ball to a smaller one. If c: [0,5] ^- X is a geodesic with c(0) = JCQ, then p is defined by r r^w

f ^('')

P^'^'^^ = \cit)

if ^ > ^ ,

if

t^r.

This gives inverse systems of maps BXQ(S) -^ Bx^ir) and Sxgis) -^ Sx^ir). Clearly, X(oo) = \imSxQ(r) and if X has extendible geodesies, then X = lim^jcoC^)- Moreover, the topologies on X and on X(oo) are those of the inverse limits. 1.42. If X is a complete CAT(0) Riemannian n-manifold, then a geodesic ray starting at XQ is uniquely determined by its unit tangent vector at XQ. It follows that X is homeomorphic to the n-disk and X(oo) to its boundary S^~K EXAMPLE

A serious problem with the above definition of the ideal boundary is that it seems to depend on the base point JCQ. A definition can be given which is independent of the choice of base point. One way to do this is to define an equivalence relation, asymptoty, on the set of unbased geodesic rays. Two such rays are asymptotic if they remain a bounded distance apart. One can then prove that the natural map {rays based at JCQ} -> {asymptoty classes of rays} is a bijection. Thus, X(oo) is the set of asymptoty classes of rays. A third description of X and X (00) in terms of "horofunctions" is given in [6]. That these definitions coincide in the Riemannian case is proved in [6]. The fact that the arguments of [6] can be extended to the general case is explained in [35] and [38] as well as in [15].

1.5. Some well-known results in geometric topology and their implications for CAT(0) spaces A closed n-manifold is homology sphere if it has the same homology as does 5". Is every homology n-sphere homeomorphic to 5"? Poincare asked this question and came up with a counterexample for n = 3. His new homology sphere M^ was S^/G where S^ is the group of quaternions of length one and where G is the binary icosahedral group (a subgroup of order 120). To distinguish M^ from S^ Poincare discovered the concept of the fundamental group and noted that 7t\ (M^) = G while 7t\(S^) = l. Later it was shown that for each n > 3 there exist homology n-spheres which are not simply connected. In fact, Kervaire showed in [48] that for n ^ 5 the fundamental group could be any finitely presented group G satisfying H\(G) = H2(G) = 0. On the other hand, the Generalized Poincare Conjecture asserts that any simply connected homology ^-sphere, n > 1, is homeomorphic to 5". For n ^ 5, this was proved by Smale, in the smooth case, (and then generalized to the PL case by Stallings and the topological case by Newman); for n = 4, it was proved by Freedman in [42].

Nonpositive curvature and reflection groups

389

Suppose C^ is a compact contractible n-manifold with boundary. Is C^ homeomorphic to the n-disk? Poincare duality implies that dC^ is a homology {n — 1)-sphere, but there is no reason for it to be simply connected. In fact, one can prove that any homology (n — 1)-sphere can be realized as the boundary of a contractible /i-manifold. (The most difficult case, n = 4, was proved in [42].) Hence, it follows from the previous paragraph that, for n > 4, there are examples where C^ is not homeomorphic to the n-disk. (The 3-dimensional Poincare Conjecture asserts that there are no such examples for n = 3.) On the other hand, for n ^ 5, the Generalized Poincare Conjecture implies that C" is homeomorphic to the n-disk if and only if dC^ is simply connected. Similarly, we can ask to what extent do open contractible ^z-manifolds differ from the interior of D" (i.e., from R"). If C^ is a compact contractible manifold, n > 2, and if dC^ is not simply connected, then the interior of C" is not homeomorphic to W. The reason is that the interior of C" is not simply connected at infinity. (Its "fundamental group at infinity" is isomorphic to 7ri(9C").) Thus, for n ^ 4 there are contractible manifolds (without boundary) which are not homeomorphic to R". This is also true for /i = 3: there is a famous example of J.H.C. Whitehead of a contractible 3-manifold which is not simply connected at infinity. In fact, the situation is more complicated than in the compact case: the end of an open contractible manifold need not be "tame", i.e., the contractible manifold might not be homeomorphic to the interior of any compact manifold with boundary. For example, the fundamental group at infinity need not be finitely generated. On the other hand, Stallings [59] (for n^5) and Freedman [42] (for n = A) showed that a contractible ^i-manifold is homeomorphic to R" if and only if it is simply connected at infinity. Here are a few questions which we shall be concerned with. QUESTION 1.43. If a contractible n-manifold admits a cocompact transformation group, then is it homeomorphic to R" ? In particular, if M" is a closed aspherical manifold, then is its universal cover homeomorphic to R'^ ?

By constructing examples from reflection groups, we shall see in Section 2.5 that the answer is "no" for each « ^ 4. This reflection group construction suggests the following question. QUESTION 1.44. Suppose M", n > 2, is a closed aspherical n-manifold with universal cover M". If the fundamental group at infinity of M" is finitely generated, then is it homeomorphic to R"? (The fundamental group at infinity 7t^(M^) is the inverse limit lim7ri(M" — K) where K ranges over the compact subsets of M".)

Something very close to an affirmative answer to this has been proved by Wright [68]. He shows that the inverse system 7t\{M^ — K) cannot be "pro-monomorphic". Next we turn to some questions involving nonpositively curved spaces. 1.45. Suppose Z" is a CATifi) manifold. (a) (Gromov) Is X^ homeomorphic to R" ? (b) If so, is (X, X{OQ)) homeomorphic to (D", S^~^)l In particular, if X^ is simply connected at infinity, then is X(oo) homeomorphic to 5"~^ ? (c) If X(oo) is a manifold, then is it homeomorphic to 5"~^ ?

QUESTION

390

M.W.Davis

The answers to all the questions in 1.45 are "no" for n > 4 and "yes" for n ^ 3. Paul Thurston [64] has proved that the answer to (a) is also "yes" when n = 4, provided the manifold has a least one "tame point," but without this hypothesis the answer is not known. The result for n = 3 is essentially due to Rolfsen [54]. For a clear discussion of these questions in dimensions 3 and 4, the reader is referred to [64]. QUESTION 1.46. If Z" is the universal cover of a nonpositively curved, closed manifold, then are the answers to Questions 1.45 the same?

A polyhedron A^" is a PL n-manifold if for each k-ctW a in A'^, Lk{a, N^) is piecewise linearly homeomorphic to S^~^~^ (where a sphere has a standard PL structure as the boundary of a convex cell). The next result, which is proved in [35], asserts that in the PL context. Questions 1.45 have affirmative answers. 1.47 (Stone [62] and [35]). Suppose X" is a piecewise Euclidean or piecewise hyperbolic polyhedron and that (a) X" is a PL manifold (i.e., the underlying polyhedral structure on X^ is that of a PL manifold), and (b) X^ is CAr(O). Then (X, X(oo)) is homeomorphic to (D", S^~^). THEOREM

The result of [62] states that X" is PL homeomorphic to R". The proof of 1.47 uses the Approximation Theorem for cell-like maps due to Siebenmann (/i ^ 5), Quinn (n = 4), Armentrout (« = 3), and Moore {n = 2). Before stating this theorem, we recall some definitions. (Our discussion is taken from [39].) A compact metrizable space C is cell-like if there is an embedding of C into the Hilbert cube / ^ so that for any neighborhood [/ of C in / ^ , the space C is contractible in U. A cell-like subspace of a manifold is cellular if it has arbitrarily small neighborhoods homeomorphic to a cell. A compact subset of S^ is pointlike if its complement is homeomorphic to M'^. A pointlike subset of 5"" is cellular in 5". A map is cell-like if each point inverse image is cell-like. 1.48 (The Approximation Theorem). Suppose f: M^ -^ N^ is a cell-like map of topological manifolds. Ifn = 3, further assume that f is cellular (i.e., each point inverse image is cellular). Then f can be approximated by a homeomorphism. THEOREM

A map which can be approximated by a homeomorphism is a near homeomorphism. We shall also need the following theorem of Brown [18]. THEOREM

1.49 (M. Brown). An inverse limit of near homeomorphisms is a near homeo-

morphism. SKETCH OF PROOF OF THEOREM 1.47. The rough idea is that one shows that the hypotheses of Theorem 1.47 imply that each infinitesimal shadow is cell-like. Since these

Nonpositive curvature and reflection groups

391

shadows are basically point inverse images of the geodesic contraction map, one can then conclude from the Approximation Theorem that each closed metric ball is homeomorphic to an n-disk and from Brown's Theorem that (X,X(oo)) = ( D " , 5 " - i ) . To be more explicit, let (r„) denote the statement of the theorem in dimension n and let {Ln) denote the following statement: If M"" has a PS structure such that (a) W is a PL manifold and (b) M"" is CAT{\), then for each r e (0,7t) and v e M^ the closed metric ball By{r) is homeomorphic to D" and By (IT) is homeomorphic to R". The inductive scheme is then (L„_i) => (L„) and (r„). For example, to see that (L„_i) =^ (7^), let X G X" and i; € Lk(x,X). By hypothesis, Lk(x,X) = S""'^ and by (L„_i), By(7t) = R"~^ Hence, by Lemma 1.43, Shad(jc, v) = Lk(x, X) — Byiix) is pointlike. In particular, since Shad(x, v) is nonempty, it follows that geodesies in X are extendible and from this that (X, X(oo)) = \im(B,,{r),

d'B,,(r))

for any base point JCQ. Moreover, using the fact that infinitesimal shadows are cell-like it is easy to produce a cell-like map from BxQ(r) to the disk of radius r in R". Further details can be found in Section 3 of [35]. D A space N is a homology n-manifold if for each x e N, Hi(N, N — x) vanishes for / ^ n and is infinite cyclic for / = n. If A^ is a polyhedron, then it is a homology n-manifold if and only if for each A:-cell a of N, Lk(a, N) has the same homology as does 5""^"^. THEOREM 1.50. Suppose X^ is a piecewise Euclidean or piecewise hyperbolic polyhedron and that (a) X" is a homology n-manifold, and (b) X" is CAr(O). Then X has extendible geodesies and its ideal boundary X{OQ) is a homology (n — 1)manifold with the same homology as S'^~^.

The point is that the arguments in the proof of Theorem 1.47 show that under the hypotheses of Theorem 1.50, each infinitesimal shadow is acyclic. 1.51. The proofs of Theorems 1.47 and 1.50 are indicative of a (conjectural) "local-to-global" principle: the topology of the ideal boundary of a CAT(0) space should be controlled by the topology of the infinitesimal shadows (reflecting the topology of the links of points). For example in Theorem 1.47, the fact that the link of each vertex of X is a PL (n — 1)-sphere is reflected in the fact that X(oo) is homeomorphic to S^~K In Theorem 1.50, the fact that the link of each vertex is a homology manifold with the same homology as S"~^ is reflected in the fact that X(oo) has the same properties. REMARK

392

M.W.Davis

Here is another result ([35, Proposition 3d.3, p. 374]) which clarifies the picture. 1.52. Suppose that X is a CAT(0) polyhedron such that the link of each vertex is a PL manifold (i.e., X has isolated PL singularities). Then any metric sphere in X, which does not pass through a vertex, is homeomorphic to the connected sum of the links of the vertices which it encloses and X (oo) is the inverse limit of this sequence of increasing connected sums. In particular, if X is a polyhedral homology manifold {so that the link of each vertex is a PL manifold and a homology sphere), then, generically, each metric sphere is a connected sum of PL homology spheres and X{OQ) is the resulting inverse limit. THEOREM

If a polyhedron is a topological manifold, then, of course, it is a homology manifold. Moreover, it can happen that the link of a ^-cell. A: > 0, in a polyhedral topological manifold can be a nonsimply connected homology sphere. This is the content of the famous Double Suspension Theorem (due to Edwards in many cases and to Cannon [19] in complete generality). This result states that if A^" is a PL manifold and a homology sphere, then its double suspension 5^ * A^" is a topological manifold (which must be homeomorphic to 5"^^ by the Generalized Poincare Conjecture). The definitive result along these lines in the following Polyhedral Manifold Characterization Theorem of [39]. 1.53 (Edwards). A polyhedral homology manifold of dimension ^ 5 is a topological manifold if and only if the link of each vertex is simply connected. THEOREM

Thus, in spite of Theorem 1.47, the possibility remains that the answers to Questions 1.45 are negative. As we shall see in Section 2.5, this is, in fact, the case (the answers to 1.45(a), (b) and (c) are negative).

1.6. Euler characteristics and the Combinatorial Gauss-Bonnet Theorem H O P E ' S CONJECTURE. Suppose M^^ is a closed Riemannian manifold, with K(M) ^ 0. Then (—l)"x(M^") ^ 0. (Here x denotes the Euler characteristic.)

1.54. (i) There is an analogous version of this conjecture for nonnegative curvature: the Euler characteristic should be nonnegative. (ii) Thurston has conjectured that Hopf's Conjecture should hold for any closed, aspherical 2n-manifold. The reason for believing this is the Gauss-Bonnet Theorem (proved by Chem in dimensions > 2). Recall that this is the following theorem. REMARK

G A U S S - B O N N E T THEOREM.

: ( M 2 ^ ) = f P.

x(

Here P is a certain 2w-form called the "Pfaffian" or the "Euler form". This leads to the following.

Nonpositive curvature and reflection groups

393

QUESTION 1.55. Does /^(M^") ^ 0 imply that ( - 1 ) " P > 0? (In other words, is ( - 1 ) " P equal to the volume form multiplied by a nonnegative function?)

In dimension 2 the answer is, of course, yes, since P is then just the volume form times the curvature. The answer is also yes in dimension 4. A proof is given by Chem in [28], where the result is attributed to Milnor. Hopf's Conjecture holds in higher dimensions under the hypothesis that the "curvature operator" is negative semi-definite (which is stronger than assuming that the sectional curvature is nonpositive). On the other hand, in dimensions ^ 6, Geroch [43] showed in 1976 that the answer to Question 1.55 is no. The following combinatorial version of the Gauss-Bonnet Theorem is a classical result. A proof can be found in [27], where one can also find a convincing argument that it is the correct analog of the smooth Gauss-Bonnet Theorem. THEOREM

1.56. Suppose X is a finite, PE cell complex. Then

X{X) =

Y.P[Lk{v,X)). V

Here P is a certain function which assigns a real number to any finite, PS cell complex. We define it below. Let or c 5"^ be a spherical ^-cell. Its dual cell a* is defined by a* = {x e S^ \d(x,a)^ 7T/2}. Let fl*(o') be the volume of a* normalized so that volume of 5^ is 1, i.e., *, . vol(a*) a*(or) = vol(5^) * If L is a finite, PS cell complex then P(L) is defined by P(L) = 1 + ^(-l)^^"^^+^^*(a), cr

where the summation is over all cells a in L. EXAMPLE 1.57. Suppose that a is an all right A:-simplex. Then a* is also an all right /:-simplex. Since S^ is tessellated by 2^"^^ copies of a* we see that a*(a) = (^)^^^ Now let L be an all right PS simplicial complex and ft the number of /-simplices in L. Then

dima+l

/

i \ /+1

The following conjecture asserts that the answer to the combinatorial version of Question 1.55 should always be yes. 1.58. Suppose that L^"~^ is a PS cell complex homeomorphic to S^"~^. If L^""-^ is CAr(l), then ( - 1 ) " P ( L 2 « - 1 ) ^ 0.

CONJECTURE

394

M.W.Davis

Thus, this conjecture impUes Hopf s Conjecture for PE manifolds. If L is a flag complex, then, by Gromov's Lemma and Example 1.57, we have the following special case. CONJECTURE

1.59. Suppose that L^""-^ is aflagcomplex which triangulates S'^''~^. Then

«-•'"('+i:(-0' '¥'• So, this conjecture implies Hopf's Conjecture for PE cubical complexes which are closed manifolds. REMARK 1.60. Conjecture 1.59 is analogous to the Lower Bound Theorem in the combinatorics (a result concerning inequalities among the ft for simplicial polytopes). For example, the Lower Bound Theorem of [67] asserts that for any simplicial complex L which triangulates S^, we have f\ > 4/o — 10. Conjecture 1.59 asserts that, if, in addition, L is a flag complex, then / i ^ 5/o — 16. Some evidence for these conjectures is provided by the following two propositions. The first result follows from recent work of Stanley [61] as was observed by Eric Babson. PROPOSITION 1.61. Suppose that L^"~^ is the barycentric subdivision of the boundary complex of a convex 2n-cell (so that L is a flag complex). Then Conjecture 1.59 holds forL.

1.62. Suppose that L^^~^ is the polar dual of a convex cell C^" in H". Then Conjecture 1.58 holds for L.

PROPOSITION

This proposition follows from a formula of Hopf (predating the general Gauss-Bonnet Theorem) which asserts the (—1)"P(L^"~^) is one-half the hyperbolic volume of C^" (suitably normalized). Further details about these conjectures and further evidence for them can be found in [22]. REMARK 1.63. A natural reaction to Conjecture 1.58 is that it might contradict Geroch's result. One could try to obtain such a contradiction as follows. Take a smooth Riemannian manifold M^" whose curvature operator at a point x is as in Geroch's result. Then try to approximate M'^^ near x by a PE cell complex with nonpositive curvature. By the main result of [27] the numbers P(L) for L a link in the complex, should approximate the Pfaffian at jc and hence, have the wrong sign. However, it is not clear that such a approximation exists. Thus, we are led to ask the following. QUESTION 1.64. Suppose M is a Riemannian manifold with K(M) ^ 0 (we could even assume the inequality is strict). Is M homeomorphic to a PE cell complex X with

K(X)^01

Nonpositive curvature and reflection groups

395

As we explained in Remark 1.37(v), the answer is yes in the constant curvature case. For our conjectures to be correct, the answer, in general, should be no. 2. Coxeter groups Coxeter groups and Coxeter systems are defined in Section 2.1. Associated to a Coxeter system there is a simplicial complex called its "nerve". The basic result of Section 2.1 is Lemma 2.6, which asserts that any finite polyhedron can occur as the nerve of some Coxeter system. Eventually, this will be used to show that Coxeter groups provide a rich and flexible source of examples. In Sections 2.2 and 2.3 we discuss a beautiful, PE cell complex E which is naturally associated to a Coxeter system {W,S). From the results of Section 1, we get the important result of Moussong (generalizing an earlier observation of Gromov), that U is nonpositively curved and hence, contractible (since it is simply connected). The connection with reflection groups is explained in Section 2.4. In Section 2.5 we briefly discuss some important special cases of this construction. 2.1. Coxeter systems DEFINITION 2.1. Let 5 be a finite set. A Coxeter matrix M = (mss') is an 5* x 5 symmetric matrix with entries in N U {oo} such that

fl ^-' = (>2 DEFINITION

if s = s\ if.^.^

2.2. Given a Coxeter matrix M, define a group W with presentation:

W = {S\ (ssT'^' = 1, V ( 5 , s ' ) e S x S). W is called a Coxeter group. If all the off-diagonal entries of M are 2 or oo, then W is called right-angled. Coxeter groups are intimately connected to the theory of reflection groups. This connection is not emphasized in this paper. For now it should suffice to mention that if a group W acts properly on a connected manifold and if W is generated by reflections (where a reflection is an involution whose fixed point set separates the manifold), then W is a Coxeter group (cf. [29, Theorem 4.1]). Given M, it is proved in [13, Chapter V, §4.3, pp. 91-92] that one can find a family (Ps)seS of linear reflections ps'.R^ -^ R^ so that ps o p^f has order mss' ^ov all (s, s^) e S X S.li follows that the map s ^^ ps extends to a representation p:W -^ GL{R^) called the canonical representation. The existence of this representation immediately implies the following: (a) the natural map 5 -> W is an injection (and henceforth, we shall identify S with its image in W),

396

M.W.Davis

(b) order {s) = 2, for all ^ G 5, (c) order {ss^) = rriss', for all {s, s^) e S x S. The pair (W, S) is called a Coxeter system. 2.3. It is proved in [13, Chapter V, §4.4, pp. 92-94] that the dual representation p*: W ^- GCCM"^)*) is faithful and has discrete image. Moreover, as explained in Section 3.1, W acts properly on certain open convex subset of (R'^)*. (These results are due to Tits.) If r is a subset of 5, then let WT denote the subgroup of W generated by T. REMARK

2.4 ([13, p. 20]). For any T C S, the pair Coxeter matrix is M\T).

LEMMA

{WT, T)

is a Coxeter system {i.e., its

Let (W, S) be a Coxeter system. We define a poset, denoted S^ {W, S) (or simply S^) by 5 ^ = {r I r c 5 and WT is finite}. It is partially ordered by inclusion. Consider S^ — {0}. It is isomorphic to the poset of simplices of an abstract simplicial complex which we shall denote hy N{W, S) (or simply A^). A^ is called the nerve of (W, S). In other words, the vertex set of N is S and a subset T of 5 spans a simplex if and only if WT is finite. EXAMPLE

2.5. If W is finite, then A^ is the simplex on S.

Which finite polyhedra occur as the nerve of some Coxeter system? The next two results show that they all do. (Compare with Corollary 1.22.) 2.6. Let L be any flag complex. Then there is a right-angled Coxeter system {W, S) with N{W, S) = L.

LEMMA

PROOF. Let S be the vertex set of L and define a Coxeter matrix (m^^O by 1 2 100

if s=s\ if {5,5^} spans an edge in L, otherwise.

If W is the associated right-angled Coxeter group, then N{W, S) = L.

D

In particular, since the barycentric subdivision of any (regular) cell complex is a flag complex, we have the following corollary. 2.7. For any finite polyhedron P, there is a right-angled Coxeter system {W, S) with N{W, S) homeomorphic to P.

COROLLARY

Nonpositive curvature and reflection groups

397

The main result of this section is the following theorem. 2.8 (Gromov, Moussong). Associated to a Coxeter system (W, S) there is a PE cell complex IJ(W, S) (= U) with the following properties. (i) The poset of cells in U is the poset ofcosets

THEOREM

WS-^ = U

W/WT.

TeSf

(ii) W acts by isometrics on E with finite stabilizers and with compact quotient. (iii) Each cell in E is simple (so that for each vertex v, Lk(v, U) is a simplicial cell complex). In fact, this complex is just N(W, S). (iv) Z is CAT(0).

2.2. Coxeter cells Throughout this subsection we suppose that W is finite. In this case, we will show that E can be identified as a convex cell in R" (n = Card(5)). The canonical representation shows that W can be represented as an orthogonal linear reflection group on R'^. The hyperplanes of reflection divide R'^ into "chambers", each of which is a simplicial cone. (See in [13, p. 85].) Choose a point x in the interior of some chamber. Define E to be the convex hull of Wx (the orbit of jc). 17 is called a Coxeter cell of type W. The proof of the next lemma is an easy exercise. LEMMA 2.9. Suppose W is finite.

(i) The vertex set of the Coxeter cell E is Wx. (ii) Each face F of E is the convex hull of a set of vertices of the form {WWT)X for some T C S and some coset WWT ofWr- (So, F is a Coxeter cell of type WT •) (iii) The poset of faces of E is therefore,

W W/WTTcS

(iv) E is simple cell. Lk(x, E) is the spherical (n — I)-simplex spanned by the outward pointing unit normals to the supporting hyperplanes of a chamber. REMARK 2.10. If x lies in a chamber with supporting hyperplanes indexed by S, then the vertex set of Lk(x, E) is naturally identified with S. Moreover, the length of the edge from 5 to ^' is TT — Tt/mss'- (In other words, the corresponding angle in a 2-cell in E is 7t -n/mss'^) EXAMPLE 2.11.

(i) IfW = Z/2, then E is an interval.

398

M.W.Davis

Fig. 4.

(ii) \fW = Dm (the dihedral group of order 2m), then i7 is a 2m-gon. (iii) If (VK, S) is the direct product of two Coxeter systems {W\, S\) and (W2, S2) (so that W = WixW2^ndS = SiU ^2), then ZiW, S) = I!(Wi, Si) x ^ ( ^ 2 , S2). In particular, if W = (Z/2)", then i7 is a n-dimensional box (= the product of n intervals), (iv) If W = Sn, the symmetric group on n letters, then Z is the {n — l)-cell called the permutahedron. The picture for n = 4 is given in Figure 4. REMARK 2.12. By choosing x to be of distance 1 from each supporting hyperplane we can normalize each Coxeter cell so that every edge length is 2.

2.3. The cell complex U (in the case where W is infinite) There is an obvious way to generalize the material of the previous subsection to the case where W is infinite. The cell complex E is defined as follows. The vertex set of Z" is W. Take a Coxeter cell of type WT for each coset WWT, T eS^. Identify the vertices of this Coxeter cell with the elements of wWj. Identify two faces of two Coxeter cells if they have the same set of vertices. This completes the definition of i7 as a cell complex. If we normalize each Coxeter cell as in Remark 2.12, then the faces of the cells are identified isometrically and hence, U has the structure of a PE cell complex. REMARK 2.13. Let A: 5 ^- (0, 00) be a function. If, in the definition of each Coxeter cell, we choose the point x to be of distance A, (5) from the hyperplane corresponding to 5, then the Coxeter cells again fit together to give a PE structure on iJ. We have arbitrarily chosen X to be the constant function. REMARK

2.14. By construction, the poset

WSf = W W/WT TeSf

Nonpositive curvature and reflection groups

399

is the poset of cells in iJ. If 7^ is any poset, then let V' be its derived complex defined as in Example 1.19 (i.e., V' is the poset of finite chains in V). V is the poset of simplices in an abstract simplicial complex. Moreover, if V is the poset of cells in a cell complex, then V' is the poset of simplices in its barycentric subdivision. Applying these remarks to the case at hand, we see that the barycentric subdivision ^' of i7 is just the geometric realization of (W5^)^ Alternatively, we could have defined iJ' as the geometric realization of i^S^)' and then remarked that the poset of chains which terminate in w^j can naturally be identified with the set of simplices in the barycentric subdivision of a Coxeter cell of type Wj. Hence, the cellulation of i7 by Coxeter cells could be recovered from i7' by collecting together the appropriate simplices. The link of a vertex. The group W acts isometrically on i7 and freely and transitively on its vertex set. Thus, there is an isometry of E which takes any vertex onto the element \ eW. What is Lki\, H) (as a simplicial complex)? A cell contains the element 1 if and only if it corresponds to some identity coset WT . Hence, the poset of simpHces in Lk{\, E) is just <S-^ - {0},i.e.,

Lk{\,i:) = N{W,S). What is the induced PS structures on Nl Two distinct vertices s and s' of A^ are connected by an edge ess' if and only if rriss' 7^ 00. By Remark 2.10, t{ess') = n — Ti/mss' (where t stands for length). Since a spherical simplex is determined by its edge lengths this determines PS structure on A^. A proof of the following lemma can be found in [13, p. 98]. LEMMA 2.15. Let T be a subset of S. Consider the T x T matrix, (Css'), where s, s' G T, and where Css' — cos(7r —n/mss')' Then Wj is finite if any only if{Css') is positive definite. PROOF. Consider the canonical representation of Wj into GL(n). Suppose Wj is finite. Then we may assume that the image of this representation is contained in 0(n). For each s e T, let Us he the outward-pointing unit normal to the hyperplane corresponding to s. Then (us - Us') = (Q^O and hence, this matrix is positive definite since (us)seT is a basis for R". Conversely, suppose that (c^^O is positive definite. Since the canonical representation preserves the corresponding bilinear form, we get that the image of this representation is contained in 0{n). Since this representation is also discrete and faithful (cf. Remark 2.3), WT is a discrete subgroup of 0(n); hence, finite. D COROLLARY

2.16. N(W,S) isa metric flag complex.

We can now prove the main result. PROOF OF THEOREM 2.8. We have already demonstrated the required properties of U except for (iv). Thus, it suffices to prove that Z is CAT(0). First of all, it is easy to see that Z is simply connected. (One argument is to observe that the 2-skeleton of U is just the universal cover of the 2-complex associated to standard presentation of W.) Hence, by

400

M.W.Davis

Theorem 1.5, it suffices to show K(Z) ^ 0. The Hnk of any vertex is isometric to A^. By Corollary 2.16 and Moussong's Lemma (Lemma 1.28) A^ is CAT(l). Therefore, K(i:)^0 (byTheoreml.il). D REMARK 2.17. Theorem 2.8 was proved by Gromov [45, pp. 131-132] in the special case where W is right-angled. The general case was proved in [51]. The point is that in the right-angled case i7 is a cubical complex so we can use Gromov's Lemma (Lemma 1.21) rather than Moussong's generaUzation of it. Combining Lemma 2.6 and Theorem 2.8 (in the right-angled case) we get the following. COROLLARY 2.18. Given a finite flag complex L, there is a finite, nonpositively curved, cubical PE complex X such that the link of each vertex in X is isomorphic to L. PROOF. Let (W, S) be the right-angled Coxeter system associated to L and let F be any torsion-free subgroup of finite index in W. (For example, F could be the kernel of the obvious epimorphism W -^ (Z/2)'^.) Set X = I!(W, S)/F. D

There is a simpler version of the above construction, which does not directly use Coxeter groups, and which gives the following slightly more general result. PROPOSITION 2.19. Let L be a finite simplicial complex. Then there is a finite, cubical PE complex X such that the link of each vertex in X is isomorphic to L. {Of course, by Lemma 1.21, L is nonpositively curved if and only if L is a flag complex.) PROOF. (Independently due to Babson and Bridson and Haefliger, [15].) Let S be the vertex set of L. Consider the cube [—1,1]*^ in the Euclidean space R'^ with standard basis {es)ses- Let X be the cubical subcomplex of [—1, l]'^ consisting of all faces parallel to a subspace of the form R^, where T is the vertex set of a simplex in L and where R^ is the subspace spanned by [es}seT' The vertex set of X is {±1}'^ and the link of each such vertex is naturally identified with L. D

2.4. Reflection groups Suppose that (W, S) is a Coxeter system, that X is a space and that {Xs)seS is a family of closed subspaces. For each x eX, let S{x) = {s e S \x eXs). Define an equivalence relation ~ on W x X by: (w;, x) ^ (w\ x') if and only if jc = x' and w~^w^ e Ws{x)' Let U(W, X) denote the quotient space (W x X)/ ^ . The group W acts on U(W, X); the orbit space is X. Moreover, we can identify X with the image of I x X inU(W,X). Thus, X is a fundamental domain for the W-action. Any translate w;X of X is called a chamber; thus, UiW, X) is decomposed into chambers. Each s in S acts on U(W, X) as a "reflection" in the following sense: the fixed point set of s separates U(W,X) into two "half-spaces" which are interchanged by 5-. (A more detailed discussion and further properties of this construction can be found in [66] or in [29].)

Nonpositive curvature and reflection groups

401

If y is a space with an action of a Coxeter group W, then W is called a reflection group on y if y is equivariantly homeomorphic io U{W, X) for some subspace X of y. EXAMPLE 2.20. Suppose that X is the cone on S and that Xs denotes the point s. Then U{W, X) is the Cayley graph of {W, S). EXAMPLE 2.21. Suppose that X is the geometric reaUzation of the poset 5 ^ of Section 2.1. Thus, a /:-simplex in X corresponds to a chain To
Here is what is going on in the above example. Given any poset V, let \V\ denote the geometric realization of its derived complex. For each p eV define subposets, V^p = {qeV\q^p}

and

V^p = {q eV \ p^

q}.

Call the subcomplexes \V^p\ faces and the \V^q\ cofaces. So we have two different decompositions of \V\, into faces or into cofaces. In the case at hand, U{W, S) = \WS^\, its faces are Coxeter cells and its cofaces are intersections of chambers.

2.5. Applications and examples (a) Two dimensional complexes. Let L be a finite graph and m an integer > 2. Let k be the girth of L (the length of the shortest circuit). If m = 2, then we assume k ^ 4. Let S = Vert(L) (the vertex set of L) and define a Coxeter matrix by ri if s=s\ f^ss' = \ m if {s,s^} spans an edge, I cx) otherwise. Let W be the resulting Coxeter group. Our assumption implies that N(W, S) = L. Thus, Z(W, S) is a. CAT(0), PE 2-complex such that each 2-cell is a regular 2m-gon and such that the link of each vertex is L. Here the condition that L was CAT (I) was just that k{7t — 7t/m) ^ 2n (which holds provided k ^ 4 if m = 2). We can give Z a piecewise hyperbolic (abbreviated PH) structure by declaring each 2-cell to be a small regular 2mgon in if-. Since the angles of such a 2m-gon will be slightly less than in the Euchdean case, we will be able to do this so that links are CAT(l) provided that k(7t — n/m) > In. This holds provided k > 4ifm = 2 and k >3ifm = 3. Thus, provided the condition holds, F can be given a PH structure which is CAT(—l). These "regular" 2-complexes can be thought of as generalization of well known examples of regular tessellations of E^ and if. Nadia Benakli has made a detailed study of these 2-complexes in her thesis [7]. For example, she shows that the ideal boundary 17 (oo) is usually a Menger curve. Benakli also

402

MM Davis

has another construction of such 2-complexes where the 2-cells are n-gons, with n odd, provided that there is a group G of automorphisms of L such that V/G is an interval {V is the barycentric subdivision). (b) Word hyperbolic Coxeter groups. In [51] Moussong also analyzed when the idea of the previous subsection (of replacing the Euclidean Coxeter cells of U by hyperbolic Coxeter cells) works in higher dimensions. Consider the following condition (*) on a Coxeter system {W,S). (*)

For any subset T of S neither of the following holds: {\)WT

= WTX X WT2 with both factors infinite,

(2) WT is a EucHdean Coxeter group with Card(r) ^ 3. Here a "Euclidean Coxeter group" means the Coxeter group of an orthogonal affine reflection group on E" with compact quotient. The "Coxeter diagrams" of these groups are listed in [13, p. 199]. 2.22 (Moussong). The following conditions are equivalent. (W, S) satisfies (*), H can be given a PH, CAT(—l) structure, W is word hyperbolic, W does not contain a subgroup isomorphic toZ-\-Z.

THEOREM

(i) (ii) (iii) (iv)

To show (i) => (ii) one wants to replace the cells of Z" by Coxeter cells in I¥^. In order for the links to remain CAT(\), one needs to know that the length of every closed geodesic in N(W, S) is strictly greater than In and that the same condition holds for the link of each simplex in N(W, S). In his proof of Lemma 1.28, Moussong analyzed exactly when a metric flag complex has closed geodesies of length equal to In. In the case at hand, it was only when conditions (1) and (2) of (*) hold. The implications (ii) ^ (iii) => (iv) ^ (i) are all either well-known or obvious. (c) Buildings. As we mentioned in Section 1.3(d) associated to each building B there is a Coxeter system (W, S) so that each "apartment" is isomorphic to a complex associated to (W, 5). Traditionally, this complex is the Coxeter complex (where each chamber is a simplex). For general Coxeter groups, however, it seems more appropriate to use the complex U\ Here Z^ denotes the barycentric subdivision of E. (In [24] we call this the "modified Coxeter complex".) This point is made by Ronan in [55, p. 184]. The buildings which arise in nature (in algebra or geometry) are usually of spherical or Euclidean type. This means that the Coxeter group W is either finite or Euclidean. In the case where W is Euclidean and irreducible and our definition agrees with the traditional one. In the general case let us agree that the correct definition of a building should be as a simplicial complex such that each apartment is isomorphic to i7^ Since, by Theorem 2.8, E' can be given the structure of a CAT(0), PE cell complex, we get an induced PE structure on the building B. As explained in Section 1.3(d)), the axioms for buildings imply this

Nonpositive curvature and reflection groups

403

structure is CAT{0). Thus, we have the following result, the details of which can be found in [32]. 2.23. Any building (correctly defined) has the structure of a PE simplicial complex which is CAr(O). THEOREM

(d) When is U a manifold? A homology manifold? Since the link of every vertex in U is isomorphic to A^, these questions can be answered as follows. PROPOSITION

2.24.

(i) E is a homology n-manifold if and only if N is a homology (n — I)-manifold with the homology of S^~^. (ii) For n ^ 5, U is a topological n-manifold if and only if N is as in (i) and N is simply connected. (iii) U is a PL n-manifold if and only if N is PL homeomorphic to S^~^. Statements (i) and (iii) are just restatements of the definitions and statement (ii) follows from Edwards' Theorem 1.52. We begin by discussing some examples of (iii) when A^ is a PL triangulation of 5"~^ EXAMPLE 2.25 (Lanner [49]). Suppose that N(W, S) is isomorphic to the boundary complex of an n-simplex. Then n = Card(5') — l,W is infinite, and for every proper subset T of S, the group WT is finite. Such groups were classified in 1950 by Lanner: they are either irreducible EucHdean reflection groups or hyperboHc reflection groups. In both cases a fundamental chamber is an n-simplex. In the Euclidean case there are a few families in each dimension n and a few exceptional cases in dimensions ^ 8. In the hyperbolic case, in dimension 2, we have the hyperbolic triangle groups: these are the groups such that Card(5) = 3 and the 3 entries p, q, r of the Coxeter matrix above the diagonal satisfy (l/p) + (l/^) + ( l / r ) < l . Furthermore, there are 9 hyperbolic examples in dimension 3, 5 more in dimension 4, and none in dimensions > 4. Complete lists can be found on pp. 133 and 199 of [13]. EXAMPLE 2.26 (Andreev [4]). Suppose that A^ is a triangulation of 5^ as a metric flag complex and that condition (*) of subsection (b) holds. Then Andreev proved that W can be realized (uniquely, up to conjugation by an isometry) as a reflection group on H^. In fact, he shows that there is a convex cell C^ in H^, the polar dual of which is N. (Thus, the faces of C^ corresponding to s and s^ make a dihedral angle of n/mss'-) W is the group generated by reflections across the faces of C^. EXAMPLE 2.27 ([65,31] and [21]). Suppose L is the boundary complex of a ndimensional octahedron (i.e., L is the n-fold join of S^ with itself). Let W be a finite Coxeter group of rank n. Use W' to label the edges of one {n — 1)-simplex in L. Label the other edges 2. This defines a Coxeter group W, with N{W,S) = L. Each chamber oiW onU is combinatorially equivalent to the cone on the dual cellulation of A^, i.e., each chamber is a combinatorial cube. If W' = (Z/2)", then each chamber actually is an n-cube.

404

M.W.Davis

In the case where W' = Sn-^\, the symmetric group, the W-manifolds Z" arise in nature. For example, there is an obvious homomorphism W ^^ W' x (Z/2)" with kernel FQ. Tomei showed in [65], the «-manifold Z/FQ can be identified with the manifold of all tridiagonal, symmetric (n -\- I) x (n -\- I) matrices with constant spectrum (of distinct eigenvalues). This is also explained in [31]. From a completely direction, it is shown in [21] that for a certain torsion-free, finite-index, normal subgroup Fi of W, I^/F\ can be identified with Gromov's "Mobius band" hyperbolization construction applied to boundary of a (n -h 1)cube, and that for a different subgroup F2, ^/F2 can be identified with the "product with interval" hyperbolization construction applied to the boundary of a simple regular (n-\-l)cell. EXAMPLE 2.28. Let L be anyflagcomplex which is a PL triangulation of 5"""^ let (W, S) be the corresponding right-angled Coxeter system with N = L. PROPOSITION 2.29. Hopf's Conjecture (from Section 1.6) for PE cubical manifolds is equivalent to Conjecture 1.59. PROOF. We saw in Section 1.6 that Conjecture 1.56 implies Hopf's Conjecture for PE cubical manifolds. Let L be an arbitrary triangulation of 5^""^ by a flag complex and let X^" be the manifold constructed in Corollary 2.18. By the Combinatorial Gauss-Bonnet Theorem (Theorem 1.56),

X{X^^) = ^ P ( L ) = [ W : r ] P ( L ) . Hence, x(^^") and P(L) have the same sign.

D

EXAMPLE 2.30. Suppose that R is a ring and that L is a flag complex which is a Rhomology (n — l)-manifold with the same homology (over R) as 5"~^ and let (W, S) be the right-angled Coxeter system with N = L. For example, we could take R = ^[;^] and L to be the lens space S^^~^/(Z/m) or the suspension of such a lens space. Then U is an /^-homology /i-manifold. Moreover, the ideal boundary i7(oo) has the same homology (over R) as does 5""^ (Compare Theorems 1.50 and 1.52). It follows that 1^ is a virtual Poincare duality group over R in the sense that any torsion-free subgroup F of finite index in W satisfies Poincare duality over R. However, as is shown in [33], if L is not an integral homology (n — 1)-sphere, then neither is E(oo). Hence, for such an L, W is not a virtual Poincare duality groups over Z.

(e) Cohomological dimension. The cohomological dimension of a torsion-free group F over a ring R, denoted by cdR(F), is nearly the same thing as the smallest dimension of an /^-acyclic CW complex on which F can act freely. (If the ring R is Z, then we omit it from our notation.) If a group G is not torsion-free, but contains a finite-index, torsion-free subgroup 7^, then the virtual cohomological dimension of G, denoted vcJ/?(G), is defined by vcdR(G) = cdR(F). If F acts freely and cocompactly on a contractible complex ^ , then cdR(F) = sup{n \ H^{Q\ R) / 0}. (See in [16, p. 209].)

Nonpositive curvature and reflection groups EXAMPLE 2.31 {Bestvina and Mess [12]). Suppose that L is a flag complex homeomorphic to MF^, that (W, S) is the right-angled Coxeter system with N = L, and that E = E(W, S). By Theorem 1.52, U(oo) is an inverse limit of connected sums of an increasing number of projective planes. In other words, i7(oo) is an inverse limit of nonorientable surfaces of increasing genus. Thus, H^(U(oo); Q) = 0 for « ^ 2, while H^(Uioo), Z) = Z/2. Consequently, H^(IJ; Q) = 0 for n ^ 3, and H^(U; Z) = Z/2. It follows that vcdqiW) = 2 and vcd(W) = 3.

2.32 (Dmnishnikov [38], Dicks and Leary [37]). There are similar examples using other Moore spaces than RP^. For example, suppose a flag complex L2 is 2-dimensional and has Hx{L2\1) = Z/3 and H/(L2;Z) = 0, for / / 1. Let iW2,S2) be the corresponding Coxeter system. Let (W\,S\) be the Coxeter system of the previous example. Put Zi = I!iW\,S\), U2 = i^(W2, ^2) and Z = Hi x 172. So, dim 17 = 6. Since Z{oo) is the join of IJ\(oo) and 1^2(00), the Kunneth formula implies that H^(i;(oo); Z) = Z/2 (g) Z/3 = 0. Thus, vcd(Wi) = vcd(W2) = 3, while vcd(W\ X W2) = 5. Hence, cohomological dimension is not additive for direct products. EXAMPLE

Since W acts on the contractible complex H with finite isotropy groups, we always have that vcd(W) ^ dim 17. As we shall see below, the inequality can be strict. Let X be a CW complex and (Xs)seS a family of subcomplexes as in Section 2.4. For any subset T of S put

XT = f]Xs seT and X^ = X. Suppose that the following two conditions hold:

(i)

XT=0,T^Sf,

(ii) XT is acyclic, T eS^. It then follows from Theorem 10.1 in [29] that U{W, X) is acyclic and, by construction, that W acts on U{W, X) with finite isotropy groups. Hence, vcd{W) ^ dimZ. In [9] Bestvina shows that, in fact, vcdiW) is equal to the smallest possible dimension of such an X. In [38] Dranishnikov gives an explicit construction of such a minimal X. (Of course, in most cases, for example if N{W, S) carries some top-dimensional homology, then vc^(W) = dimi:.) (f) The Eilenberg-Ganea Problem. The geometric dimension of a torsion-free group F, denotes gd(r), is the smallest dimension of a K(r, 1) complex. Equivalently, it is the smallest dimension of a contractible CW complex on which F can act freely. The following result is proved in [40] when cd(F) / 1 and in [60] in the case cd(F) = 1. 2.33 (Eilenberg and Ganea, Stallings). (i) Ifcd(F) i^ 2, then cd(F) = gd(F). (ii) Ifcd(F) = 2, then either gd(F) = 2 or gd(F) = 3.

THEOREM

The Eilenberg-Ganea Problem is the question of whether or not there exists a group F with cd(F) = 2 and gd(F) = 3.

405

406

M.W.Davis

Let L be 2-dimensional flag complex which is (a) acyclic and (b) not simply connected. Let (W, S) be the corresponding right-angled Coxeter system with N = L.lf X denotes the geometric reahzation of S^ then, as in Example 2.21, U(W, X) = U. Let XQ be the geometric realization of Si^^ (so that XQ is the barycentric subdivision of L and X is the cone on X) and put XQS = Xs and ZQ = U(W, XQ) . By the last paragraph of subsection (e), Eo is an acyclic 2-complex. It follows that for any torsion-free subgroup F of finite index in W, cd(r) = 2. On the other hand, the only natural contractible complex on which F acts is i7, which has dimension 3. This leads to the following CONJECTURE 2.34 (Bestvina). For any F as above, gdiF) = 3. The reason for believing this is that it seems that XQ should embed in the universal cover EF of any K(F, 1). Furthermore, since TTI (XQ) is not trivial it should be impossible to embed it in any contractible 2-complex, since it should not be possible to kill Tt\ (XQ) by adding the same number of 1- and 2-cells. (This is related to the Kervaire Conjecture: if G is a nontrivial group, then any group obtained from G by adding one generator and one relation is also nontrivial.) (g) Is U homeomorphic to W ? PROPOSITION

2.35.

(i) IfN is a PL triangulation ofS^~K then E is PL homeomorphic to W^. (ii) If N is a PL homology sphere {i.e., N is a PL manifold) and 7i\ (N) is not trivial, then Z is not simply connected at infinity. (However, E is not a manifold, rather it is only a homology manifold.) (iii) If N is a simply connected homology manifold with the homology of S^~^, then, for n^ 5, U is a contractible manifold. PROOF. Statement (i) follows from Stone's result (Theorem 1.47), statement (ii) from Proposition 1.52, and statement (iii) from Edwards' Theorem 1.53. D

We are now in position to tackle Questions 1.45 and 1.46. 2.36 ([35, p. 383]). For each n^5, that the corresponding PE polyhedron U is (a) CAT{0), (b) a topological n-manifold, and (c) not homeomorphic to R".

PROPOSITION

there is a Coxeter system (W, S) so

PROOF. Let A^~^ be a compact acycUc PL manifold with boundary such that (1) 7i\(dA) -^ 7t\{A) is onto and (2) the double of A along dA is not simply connected. Take a triangulation of A as a flag complex so that 3 A is a full subcomplex. Let A^ be the simplicial complex resulting from attaching the cone on dA to A. Let (W, S) be the right-angled Coxeter system with N(W, S) = N. By (1), A^ is simply connected, so by

Nonpositive curvature and reflection groups

407

Proposition 2.35(iii), iJ is a topological n-manifold. Let {W\, 5'i) be the right-angled Coxeter system whose nerve is the double of A along dA.li follows from (2) and part (ii) of Proposition 2.35 that the resulting contractible complex is not simply connected at infinity. On the other hand, W\ can be identified with an index two subgroup of W. (Double the fundamental chamber X of U along Xs where s corresponds to the cone point in A^.) Since the fundamental group at infinity of 17 depends only on W (or W\) we see that U is also not simply connected at infinity and hence, not homeomorphic to R". D In [2] it is proved that in the case where N{W,S)i^di nonsimply connected PL homology sphere (as in Proposition 2.35(ii)), a modified version of U can be taken to be a topological manifold. The idea is similar to that in the previous proof: one blows up the PL singularities of E from isolated vertices into intervals. More precisely, we have the following result. 2.37 ([2]). Let L^~^ be a PL homology sphere. Then there is a right-angled Coxeter system {W, S), with N(W, S) = L, and a PE cubical complex E\ with W-action such that (a) Ei is CAT(0) and (b) U\ is a topological n-manifold. PROPOSITION

IDEA OF PROOF. We can find a codimension-one homology sphere LQ C L such that (1) Lo divides L into two pieces L\ and L2 (each of which is an acyclic manifold with boundary) and (2) 7T\ (LQ) —> 7T\ (Li) is onto, for / = 1, 2. Triangulate L as a flag complex so that Lo is a full subcomplex and let (W, S) be the right-angled Coxeter system such that N(W, S) = L. For / = 1,2, let Nt denote the union of L/ with the cone over LQ. Then Ni is simply connected. In the construction of 17 a fundamental chamber is essentially a cubical cone over L. As explained in [2], to construct U\, one uses as a chamber the union of two cubical cones: one over A^i and the other over A^2. These cones are glued together along the cone on LQ. Such a chamber now has PL singularities along an interval which connects the two cone points c\ and C2. The hnk of c/ is Ni, which is simply connected; hence, by Edwards' Theorem 1.53, U\ is a topological manifold. D

2.38. In the previous two propositions we have shown that Questions 1.45(a) and 1.46(a) have negative answers. As for part (b), let U\ be as in the previous proposition and consider the CAT(0) manifold X = Ui x R. The group W xZ acts on X with compact quotient. Since X is simply connected at infinity, it is homeomorphic to R"~^^ (by [59]). On the other hand, X(oo) is the join of U\ (00) and S^; hence, it is not homeomorphic to 5". Question 1.45(c) asks if the ideal boundary is a manifold, then must it be a sphere? It is proved in [3] that the answer is no. The construction is very similar to the one in the above proof of Proposition 2.37. Indeed, if A^i and N2 are as in the above proof, then one takes two Euclidean cones (or two hyperbolic cones) on 5i and N2 and glue them together along the cone on LQ. The resulting space Y^ is a complete CAT{0) manifold (it is CAT(—l) if we use hyperbolic cones) and Y(oc) is the original PL homology sphere L"~^ On the other hand, it seems unlikely (because of [68]) that one could construct such an example with a cocompact group of isometrics. Thus, the answer to Question 1.46(c) is probably "yes". REMARK

408

M.W.Davis

3. Artin groups As explained in Section 3.1 any Coxeter group (finite or not) has a representation as a reflection group on a real vector space. Take the complexification of this vector space. It contains a certain convex open subset such that after deleting the reflection hyperplanes, we obtain an open manifold M on which the Coxeter group W acts freely. The fundamental group of M/ W is the "Artin group" A associated to W. When W is finite, Deligne proved that M/ W is a ^ ( A , l)-space. The conjecture that this should always be the case, is here called the "Main Conjecture". The purpose of this chapter is to outline some work on this conjecture in [24] and [25]. Associated to the Artin group there is a cell complex 0 (which is very similar to i7). It turns out (Corollary 3.30) that proving the Main Conjecture for W is equivalent to showing 0 is contractible. The complex 0 has a natural PE structure, which we conjecture is always CAT(0). We do not know how to prove this; however, in Section 3.5 we show that there is a (less natural) cubical structure on 0 and that in "most cases" it is CAT(0). Hence, the Main Conjecture holds in most cases. 3.1. Hyperplane complements Let Sn denote the symmetric group on n letters. It acts on R" by permutation of coordinates. In fact, this action is as an orthogonal reflection group: the reflections are the transpositions (ij), 1 ^ / < 7 ^ n, the corresponding reflection hyperplanes are the Htj = {jc G M" | JC/ = Xj}. Complexifying we get an action of Sn on C" = R" 0 C such that Sn acts freely on

M = e-|J(/f/y0C). Thus, M/Sn is the configuration space of unordered sets of n distinct points in C. It is a classical fact that the fundamental group of M/Sn is Bn, the braid group on n strands. The following result is also classical. 3.1 (Fox and Neuwirth [41]). (1) M is a K(7t,l) space, where n is PBn the pure braid group {i.e., PBn is the kernel ofBn-^Sn). (2) M/Sn is a K(7t,l) space, for n = Bn.

THEOREM

Next suppose that W is a finite reflection group on R" and that

M = C"-lJ//,0C, r

where the union is over all reflections r in W (i.e., all conjugates of elements in S) and where Hr is the hyperplane fixed by r. Arnold and Brieskom asked if the analogous result to Theorem 3.1 holds in this context. In [36], Dehgne proved that this was indeed the case. THEOREM 3.2 (Deligne [36]). Suppose that W is a finite reflection group. Then M/W is a K(7t,l) space, where ir is the ''Artin group" associated to W (as defined below).

Nonpositive curvature and reflection groups

409

Artin groups. Suppose that (VK, S) is a Coxeter system and that M = {russ') is the associated Coxeter matrix. Introduce a new set of symbols X = {xs \s e S}, one for each element of ^. NOTATION. If m is an integer ^ 2, then let prod(x, >'; m) denote the word: xyx • • •, where there are a total of m in letters in a word. DEFINITION 3.3. The Artin group associated to (W, S) (or to M) is the group generated by X and with relations:

prod(x5, x^/; niss') = prod(x^s x^; m^^/), where {s, s') range over all elements of S x S such that s ^s^ and rriss' ^ oo. REMARK 3.4. If we add the relations (xs)^ = 1, then the relation appearing in the previous definition can be rewritten as (x^x^O'""' = 1; hence, we recover the standard presentation of W. Thus, if A is the Artin group associated to (W, S), we see that there is a canonical surjection p : A^^ W which send jc^ to ^. EXAMPLE

3.5. If W is 5^, then the associated Artin group is 5„.

It is natural to ask if Theorem 3.2 holds in the case where W is infinite. In order to make sense of this question we first need to discuss what is meant by a "linear reflection group" in the infinite case. Linear reflection groups. Let V be a finite dimensional real vector space. A linear reflection on V means a linear involution with fixed space a hyperplane. Suppose that C is a convex polyhedral cone in V (Figure 5) and that 5" is a finite set which indexes the set of codimension-one faces of C. Thus, {Cs)ses will be the family of codimension-one faces of C. Let Hs denote the linear hyperplane spanned byC,. For each s e S, choose a reflection ps with fixed subspace Hs. Let W denote the subgroup of GLiV) generated by {ps \s eS}.

Fig. 5.

410

M.W. Davis

DEFINITION 3.6. W is a linear reflection group if wC n C = id or all w e W, w ^ I. (Here C denotes the interior of C.) DEFINITION 3.7.

Let

7= y wc wCW

and let / denote the interior of / . / is called the Tits cone. 3.8. Consider the quadratic form model of H^: the hyperboUc plane is identified with one sheet of a hyperboloid in R^' ^ (3-dimensional Minkowski space). An isometric reflection on H^ extends to a linear reflection on E^' * preserving the indefinite quadratic form. Now suppose that W is the reflection group on H^ generated by the reflections across the edges of a hyperbolic polygon with angles of the form Jt/m. Then W can be regarded as a linear reflection group on R^' ^. (See Figure 6.) EXAMPLE

In this case the interior / of the Tits cone is just the interior of the light cone. THEOREM 3.9 (Vinberg [66]). Suppose that W C GL{V) is a linear reflection group with fundamental polyhedral cone C. Put C-^ = {x eC \Wx is finite). Then (i) {W, S) is a Coxeter system, (ii) / is a convex cone, (iii) / is W-stable and W acts properly on it,

(iv) inc = cf, (v) theposet offace ofC^ is S-f {where S-^ = {T C S \ Wj is finite}).

Fig. 6.

Nonpositive curvature and reflection groups

411

Let W be as in Vinberg's Theorem and consider the domain V + i/ in V (g) C (V -f i/ denotes the set of vectors whose imaginary part Hes in / ) . Set

M = (y+i/)-|J(//,0C). The following is the main conjecture which we shall be concerned with in this chapter. According to [50] it is due to Arnold, Pham and Thom. CONJECTURE 3.10 {The ''Main Conjecture''). M/ W is a K{7t, 1) space, where n = Aw, the Artin group associated to {W,S).

Some progress on this was made in the thesis of van der Lek [50], where the following result is proved. (Another proof can be found in [24].) PROPOSITION 3.11

(van der Lek). 7Z\ {M/ W) = Aw-

The Main Conjecture can also be formulated in terms of the cell complex U which was introduced in the previous chapter in Section 2.3. In fact, in view of Theorem 3.9(v), the following lemma is not surprising. LEMMA 3.12 ([24, Section 1.2]). There are W-isovariant homotopy equivalences: I--E

and

V -\-iI -- I x I -- U x U.

Set

Y = (i: X

Z)-[JZrXi;r, r

where Ur denotes the subcomplex of I!' fixed by r. Then

Y/W--M/W. Hence, we have the following CONJECTURE

3.13 {Reformulation of the Main Conjecture). Y/W i^di K{Aw A)-

3.2. The Salvetti complex In this section, which is independent of the last three sections of this chapter, we describe a PE cell complex H homotopy equivalent to M. The quotient space U/W is3.finiteCW complex. Hence, when the Main Conjecture holds, U/W will be a ^(A^^, 1) space. The complete details of this construction are given in [25].

412

M.W.Davis

In Section 2.1, we considered two posets: S^ = [T CS\WT\S

finite}

and

WS^ = ] J

W/WT.

TeSf

Here we consider a third poset W x S-^. The partial ordering on W x S^ is defined as follows: (if, T) < (w\ T') if and only if (i) T < T\ (ii) w'^w^ e WT', and (iii) for all t eT, liw'^w^) < i(tw~^w') (where i denotes word length in W). There is a natural projection n :W x S-^^ ^^ WS-^ defined by (w, T) -^ WWT. Conditions (i) and (ii) just mean that n is order-preserving. Condition (iii) comes out of the proof of Proposition 3.14, below. The quickest way to define U is to first define its barycentric subdivision Z"; it is the geometric realization of the derived complex of W x S^. One then observes that the union of simplices with maximal vertex is (w,T) can be identified with a Coxeter cell of type WT • If Z is a cell complex then ViZ) denotes the poset of cells in Z. For example, V(U) =

wsf. 3.14 (Salvetti [56], Chamey and Davis [25]). There is a PE cell complex i such that (i) V{i:) = W X 5^, (ii) each cell of E is a Coxeter cell, (iii) W acts freely on E, (iv) 5 is W-equivariantly homotopy equivalent to M (or to Y) and hence, Z/W ^ M/W.

PROPOSITION

SKETCH OF PROOF. First, for each (w,T) in W x S-^ we will describe two open sets in I7^ Let U^. j . denote the open star of the vertex corresponding to WWT in iJ^ Let ^(w T) d^^ote the intersection of the open "half spaces" in E^ which are bounded by the Er with r a reflection in WWTW~^ and which contain the vertex w. (U^/ ^^x is an open "sector".) We note that Ui^^ j . contains no point in U^. j . with nontrivial isotropy group. Consider Y = {IlxE) — l j ( ^ r x Ur)- Let U(^u)j) = U'^^ j^ x U'^^ jy One checks easily that (a) U(w,T) C F, (b) {U(u)j)} is an open cover of Y, (c) each nonempty intersection of elements in this cover is contractible, and (d) the nerve of this cover is U^ (the geometric reahzation of (W x S^Y). The proposition follows. D REMARK. In [56] Salvetti carries out the above construction for arbitrary hyperplane complements. The special case above is done in [25]. When W is the symmetric group, the result was known earlier (for example, to J. Milgram and C. Squier).

The CW complex L/W has one cell of dimension Card(r) for each T eS^ .In particular, when W is finite, E/W is the CW complex formed by identifying faces of a single

Nonpositive curvature and reflection groups

413

b a

Fig. 7.

Coxeter cell: the precise identifications can be worked out using conditions (i), (ii) and (iii) in the definition of the partial order on W x <S^. COROLLARY

3.15. The Main Conjecture holds if and only if U /W is a K{Aw, 1) space.

3.16. The Main Conjecture holds when W is finite. In particular, E/W is a K{A, 1) when A is an Artin group. For example, when A = ^ 3 , a K(B3,1) can be constructed by identifying edges of a hexagon in the pattern shown in Figure 7. EXAMPLE

EXAMPLE 3.17. lfW = (Z/2)'', then E/ W can be identified with the «-torus T"" with its usual cell structure (obtained by identifying opposite faces of the n-cubc). EXAMPLE 3.18. Suppose W is right-angled. The Main conjecture holds for W (see Theorem 3.35 below). In this case Z/W can be identified with a subcomplex of T", n = Card(5), as follows. Let (Zs)seS be the standard linear coordinates on T^. Given a point z = (Zs)seS in T" define the support of z by Supp(z) = {s e S \ Zs ^0 (mod 1)}. Then Z/ W can be identified with the subcomplex of T" consisting of all z€T^ such that Supp(z) is the vertex set of a simplex in N(W, S). Thus, Z/W is a union of subtori, one for each simplex in N(W, S).

In [11], Bestvina and Brady made use of such "right-angled Artin groups" to construct some startling examples of non-finitely presented groups. Suppose that N(W, S) is a finite acyclic complex which is not simply connected (as in 2.5(f)). Let A be the associated Artin group and (^: A -> Z the homomorphism which sends each generator Xs to a generator of Z and let H denote the kernel of (p. Let f: Z/W ^^ S^ be the restriction of the linear map T" -^ S^ which sends z to Zzs (mod 1). Then / induces cp onni. It is proved in [11] that (a) H is not finitely presented and (b) for any 0 e S\ the inverse image of f~^ (0) in the universal cover of Z/W is acyclic. Such H are the first examples of torsion-free groups which are (a) not finitely presented and (b) act freely and cocompactly on acyclic complexes. 3.19 ([25]). Suppose the Main Conjecture holds for (W, S). Then (i) cd(Aw) = dimZ = dimZ. (ii) x(Aw) = X(S/W) = l-xiNiW,S)).

COROLLARY

414

M.W.Davis

Lk(vX) Fig. 8.

A naive idea for proving the Main Conjecture would be to show that K(Z) < 0. This actually works when W is right-angled (as is proved in [25]); moreover, this fact plays a key role in the work of Bestvina and Brady. However, it does not work in general. For example, when A = B3, the link of a vertex in the complex in Example 3.16 is the graph shown in Figure 8. If each edge length is 27r/3, but then the digons have length 47r/3 which is <2n.

3.3. Complexes of groups Graphs of groups. We begin by recalling some well-known results from the theory of graphs of groups (cf. [57]). Let ^ be a graph, V{Q) the poset of cells in ^ and ViQY^ the dual poset, thought of as a category. A graph of groups over ^ is a functor Q from V{QyP to the category of groups and monomorphisms. Thus, to each vertex i; of ^ we are given a group Q{v) and similarly a group Q{e) for each edge e. Moreover, if i; is a vertex of e, then there is a monomorphism Q{e) -^ Q{v). From these data one can construct a group G, called ihc fundamental group ofQ and denoted by n\ {Q). The basic result in the theory is the following. 3.20 ([57]). Given a graph of groups Q over Q, there exists a tree T with G-action (G = 7t\ (Q)) so that the following hold. (i) T/G = Q. (ii) Suppose e is an edge ofQ, v a vertex ofe, e a lift ofetoT and v the corresponding vertex ofe. Then there is an isomorphism Gy=Q(v) taking G^ onto the image of Q{e).

THEOREM

One consequence of (ii) is that the natural map Q{v) ^^ 7X\ {Q) is injective (since it is isomorphic to the inclusion Gy CG). In the language of [46] this means that G is developable. The tree T is called the universal cover of Q. It is unique up to G-isomorphism. The other feature of a graph of groups is that this universal cover is not only simply connected, it is contractible. An important application of this theory is to the problem of gluing together various K{7t,l) spaces and then being able to decide if the result is also aspherical.

Nonpositive curvature and reflection groups

415

3.21. An aspherical realization of a graph of groups ^ is a CW complex B and a map p\B ^^ Q^ so that for each vertex v of Q, /7~^ (Star(i;)) is a ^ ( ^ ( f ) , 1). Here "Star" refers to the open star of a vertex in the barycentric subdivision Q'. (Actually, this is only an approximation of the correct definition which can be found in [47, Sections 3.3 and 3.4].) DEHNITION

REMARK 3.22. It is proved in [47], in the more general context of complexes of groups, that aspherical reaUzations exist and are unique up to homotopy.

We will usually denote an aspherical realization oi Q by BQ and call it the classifying space of ^. As a definition of TTI {Q) we could take the usual fundamental group of BQ. The following result is classical; its proof goes back to J.H.C. Whitehead. THEOREM

7Ti(g).

3.23. Let Q be a graph of groups, BQ an aspherical realization, and G = ThenBgisaK(G,l).

PROOF. Let EG be the universal cover of K(G, 1). Consider the diagonal G-action on EG X T. Projection on the second factor EG x T -> T induces a map of quotient spaces EG XG T ^^ Q which is clearly an aspherical realization. Hence, we can take BQ = EG XG 7". On the other hand, the universal cover of EG XQ T is EG x T which is contractible. D

Complexes of groups. Here we present a simplified version of the theory developed in [46] and [47]. (For the applications we have in mind we do not need the most general version of the theory.) DEFINITION 3.24. Let P be a poset. A simple complex of groups over P is a functor Q from V to the category of groups and monomorphisms. REMARK. In the general situation of [47], P need not be a poset but only a "category without loop". More importantly, Q need not be a functor. The appropriate triangular diagrams relating compositions of morphisms need not commute on the nose but only up to conjugation by some elements in the target group; furthermore, these elements must be kept track of.

The concept of an "aspherical realization" is defined as before. Such an aspherical reaUzation is devoted by BQ and called the classifying space of Q. By definition, 7t\(Q) = 7t\(BQ). It can also be defined via generators and relations [47, §12.8]. A complex of groups need not be developable. Moreover, even if it is developable its universal cover need not be contractible. EXAMPLE 3.25. Suppose V = V{A^yP where A^ is a 2-simplex. Define a complex of groups Q by Figure 9, where Dm denotes the dihedral groups of order 2m. Then 7Z\ (Q) is a Coxeter group W on three generators. Q is developable. Assume that (l/p) + (l/q) + (1/r) > 1. Then W is finite. The universal cover of Q is homeomorphic to 5^ (triangulated

416

M.W.Davis

Fig. 9.

as the Coxeter complex) and BQ = EW xw S^ which is not a K(W, 1) (its universal cover is homotopy equivalent to 5^). What is missing in higher dimensions is a hypothesis of nonpositive curvature (which is automatic in the case of a graph of groups). The proof of the following basic result is outlined by Haefliger in [46] and some of the details are supplied in the thesis of Spieler [58]. 3.26 (Haefliger, Spieler). Let Q bea complex of groups. Suppose that Q admits "a metric with K ^0". Then (i) Q is developable, (ii) its universal cover is CAr(O) {and hence, contractible), (iii) BQ is a K(G, 1) space.

THEOREM

REMARK. Suppose ^ is a complex of groups over a poset V. The hypothesis of nonpositive curvature means that the geometric realization of V^ admits a PE "orbihedral structure" with jRT < 0. In other words, the local models of a universal cover must be PE and CAT(0).

3.27. Suppose (W, S) is a Coxeter system and that «S^ is the poset defined in Section 2.1. Define a simple complex of groups W over «S^ to be the functor W(T) = WT . Then it is easily seen that W is developable and that 7t\ (W) = W. Moreover, its universal cover is just the geometric realization of the poset WS-^, i.e., it is E^ (the barycentric subdivision of E). Moussong's Theorem 2.8 shows that the natural PE structure on U^ is CAr(O), i.e., we are in the situation of Theorem 3.26. EXAMPLE

EXAMPLE 3.28. Let A be the Artin group associated to (W, S). Let S-^ be as above. Define a simple complex of groups A over S-^ to be the functor A(T) = Aj, where AT is the Artin group corresponding to (WT, T) (AT is an Artin group of "finite type"). It follows easily that TTI (.4) = A. Moreover, it can be shown that for each T eS^,AT -^ A is injective, i.e., A is developable. Consider the set,

A5^= ]J A/AT, TeSf

Nonpositive curvature and reflection groups

417

partially ordered by inclusion. The geometric realization

3.4. Reinterpretation of the Main Conjecture Recall that

r = (i;x i;)-|Ji:, xr,. Let X denote the geometric realization of S^. Let 7t \Y/W ^^ U' /W = X be the map induced by projection on the first factor. PROPOSITION

3.29 ([25]). n :Y/W ^^ X is an aspherical realization of A (from Exam-

ple 3.28). PROOF. Basically, this is just what Deligne's Theorem (Theorem 3.2) tells us. Indeed, if T is a vertex of X, then

7r"^(Star(r)) = rStar(l, T) x E - (Jhyperplanesl/l^r which is homotopy equivalent the orbit space of the hyperplane complement for the finite Coxeter group WT- By Deligne's result, this is a A'(A7,1). Thus, in the general situation, Y/ W is homotopy equivalent to BA. D COROLLARY

3.30. The Main Conjecture holds for (W, S) if and only if0 is contractible.

PROOF. We are using the form of the Main Conjecture in 3.13. By Proposition 3.29, Y/ W is homotopy equivalent to EAx A 0- Therefore, the universal cover of Y/ W is contractible if and only if 0 is contractible. D

0 is ''building-like'\ As explained in Section 3.1, there is a natural epimorphism p : A^^W. We can define a section (p:W ^^ A of p SLS follows. Given w in W, write w = s\ " -Sm where s\" -Sm isa word of minimum length for w. Set (p(w) = JC^, • • • Xs^ G A. It can be shown that (p is well-defined (i.e., the value of cpiw) does not depend on the choice of minimal word). Of course, cp is not a homomorphism. The map (p induces a embedding of posets WS^ -^ AS-f and therefore, a simplicial embedding E' -^ 0. The translates of U' by elements of A are the apartment-like subcomplexes. We have that

0=[Jai:' a£A

418

M.W.Davis

and in this sense, 0 is "building-like". 0 is not in fact a building: two points need not lie in a common apartment. Nevertheless, there is an obvious idea for trying to prove 0 is contractible. Give 0 a PE structure by declaring each apartment-like complex to be isometric to E^ with its natural PE structure (described in Sections 2.2 and 2.3). Then prove 0 is CAT(0). To attack this we must study links of vertices in 0. The simplicial complexes U and 0. Suppose for the moment that W is a finite Coxeter group. Let Z\ be a simplex, the codimension-one faces of which are indexed by S.ifs e S, then As denotes the corresponding face. Given x e A, put ^(JC) = {^ 6 5 | JC G AS). Define Zw = (W X A)/--, where the equivalence relation ~ is defined by {w,x) ^ (w\ x') if and only if jc = x' and u;~^ if^ € Ws(x)' ^w is the usual Coxeter complex of W. Its poset of simplices is

1]JW/WT]

.

It can be identified with the triangulation of the unit sphere in R" where the (n — 1) simplices are the intersection of 5"~^ with the translates of a fundamental simplicial cone. In other words, if we identify A with a spherical simplex so that the dihedral angle along As n As^ is n/mss', then the induced PS structure on Ew is that of a round sphere. Similarly, define 0w = (Ax Z\)/~, where ~ is defined by (a,x) ^ (a^ JCO if and only if x = s' and a~^a^ G As(x)- The simplicial complex 0w is called the Deligne complex of {W,S). Deligne proved, in [36], that 0w is homotopy equivalent to a wedge of {n — 1)-spheres (where n — \= dim A). As in the previous subsection.

0w=

\^aEw' aeA

So, 0vy is spherical-building-like. If we identify A with a spherical simplex as above, then each apartment-like subcomplex aI7vy is isometric to a round sphere. Links of vertices in 0. The vertices of 0 correspond to elements of A<S^, i.e., to cosets of the form aAj.T eS^. We classify these into three types. Type 1. r = 0. In this case Lk{v, 0) = N{W, S) (the same link as for a vertex of E). Type 2. T is maximal in <S^. In this case, Lk{v, 0) = 0WT' (^^ the analogous case for U^ the link would be a round sphere.) Type 3. In the general case, Lk(v, 0 ) is the orthogonal join of a link of a simplex in N(W, S) and a link of Type 2.

Nonpositive curvature and reflection groups

419

Thus, if we give 0 its natural PE structure every link is of the form N{W,S) (which is CAT{\) by Moussong's Lemma and Corollary 2.16), or a Deligne complex associated to a finite Coxeter group, or a join of these two types. Thus,

3.32. Conjecture 3.31 implies the Main Conjecture 3.10.

It follows from a lemma of [5, p. 210] that Conjecture 3.31 holds when W = Dm, the dihedral group of order 2m. (The lenmia asserts that (Pom has no circuits of length < 2m.) This yields the following. COROLLARY

3.33 ([25]). The Main Conjecture holds whenever dim

3.5. A cubical structure on

3.34. Let W be a finite Coxeter group. Then the Deligne complex 0w is a flag

complex.

Fig. 10.

420

M.W.Davis

For vertices of Type 1 the link is again N(W, S). The conclusion is that the cubical structure on 0 is CAT(0) exactly when the cubical structure on U is CAT(0). THEOREM

3.35 ([25]). The Main Conjecture is true when N{W, S) is a flag complex.

REMARK 3.36. Taken together, Corollary 3.33 and Theorem 3.35 constitute a proof of the Main Conjecture in most cases.

3.37. In [49], Lanner showed that N{WT, T) is the boundary of a simplex (an "empty simplex"in N{W,S)) if and only if WT is a reflection group on either hyperbolic space or Euclidean space with fundamental chamber a simplex. (See Example 2.25.) Hence, N{W, S) is a flag complex if and only if for all subsets T of 5, with Card(r) ^ 3, neither of the following conditions hold: (a) WT is an irreducible, affine Euclidean reflection group, (b) WT is a hyperbolic reflection group with fundamental chamber a simplex. REMARK

References [1] A.D. Aleksandrov, V.N. Berestovskii and I. Nikolaev, Generalized Riemannian spaces, Russian Math. Surveys 41 (1986), 1-54. [2] F. Ancel, M. Davis and C. Guilbault, CAr(O) reflection manifolds. Geometric Topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math. 2.1, Amer. Math. Soc, Providence, RI (1997), pp. 441-445. [3] F. Ancel and C. Guilbault, Interiors of compact contractible N-manifolds are hyperbolic (N ^ 5), J. Differential Geom. 45 (1997), 1-32. [4] E.M. Andreev, On convex polyhedra in Lobacevskii spaces. Math. USSR Sbomik 10 (1970), 413^40. [5] K.I. Appel and P.E. Schupp, Artin groups and infinite Coxeter groups. Invent. Math. 72 (1983), 201-220. [6] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Math., Vol. 61, Birkhauser, Stuttgart (1985). [7] N. Benakli, Polyedres hyperboliques passage du local au global. These, Universite de Paris-Sud (1992). [8] V.N. Berestovskii, Borsuk's problem on the metrization of a polyhedron, Soviet Math. Dokl. 27 (1983), 56-59. [9] M. Bestvina, The virtual cohomological dimension of Coxeter groups. Geometric Group Theory I, G. Niblo and M. Roller, eds, Cambridge University Press, Cambridge (1993). [10] M. Bestvina, The local homology properties of boundaries of groups, Michigan Math. J. 43 (1996), 123129. [11] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), 123-129. [12] M. Bestvina and G. Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), 469481. [13] N. Bourbaki, Groupes et Algebres de Lie, Chapters IV-VI, Masson, Paris (1981). [14] M.R. Bridson, Geodesies and curature in metric simplicial complexes. Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger and A. Verjovsky, eds. World Scientific, Singapore (1991), 373^63. [15] M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer, New York (1999). [16] K. Brown, Cohomology of Groups, Springer, New York (1982). [17] K. Brown, Buildings, Springer, New York (1989). [18] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478^81. [19] J.W. Cannon, Shrinking cell-like decompositions of manifolds: codimension three, Ann. of Math. 110 (1979), 83-112.

Nonpositive curvature and reflection groups

421

[20] R. Chamey and M.W. Davis, Singular metrics of nonpositive curvature on branched covers ofRiemannian manifolds, Amer. J. Math. 115 (1993), 929-1009. [21] R. Chamey and M.W. Davis, Strict hyperbolization. Topology 34 (1995), 329-350. [22] R. Chamey and M.W. Davis, The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold. Pacific J. Math. 171 (1995), 117-137. [23] R. Chamey and M.W. Davis, The polar dual of a convex polyhedral set in hyperbolic space, Michigan Math. J. 42 (1995), 479-510. [24] R. Chamey and M.W. Davis, The K{R, \)-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995), 597-627. [25] R. Chamey and M.W. Davis, Finite K{TT, 1) 'sforArtin groups. Prospects in Topology, F. Quinn, ed.. Annals of Math. Studies, Vol. 138, Princeton Univ. Press, Princeton (1995), 110-124. [26] R. Chamey, M. Davis and G. Moussong, Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds, Michigan Math. J. 44 (1997), 201-208. [27] J. Cheeger, W. Miiller and R. Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), 405^54. [28] S.S. Chem, On curvature and characteristic classes of a Riemannian manifold, Abh. Math. Sem. Univ. Hamburg 20 (1956), 117-126. [29] M.W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. (2) 117 (1983), 293-325. [30] M.W. Davis, Coxeter groups and aspherical manifolds. Algebraic Topology Aarhus 1982, Lecture Notes in Math., Vol. 1051, Springer, New York (1984), 197-221. [31] M.W. Davis, Some aspherical manifolds, Duke Math. J. 55 (1987), 105-139. [32] M.W. Davis, Buildings are CAT{0), Geometry and Cohomology in Group Theory, P. KrophoUer, G. Niblo and R. Stohr, eds, London Math. Soc. Lecture Notes, Vol. 252, Cambridge Univ. Press, Cambridge (1998), 108-123. [33] M.W. Davis, The cohomology of a Coxeter group with group ring coefficients, Duke Math. J. 91 (1998), 297-314. [34] M.W. Davis and J.-C. Hausmann, Aspherical manifolds without smooth or PL structure. Algebraic Topology, Lecture Notes in Math., Vol. 1370, Springer, New York (1989). [35] M.W. Davis and T. Januszkiewicz, Hyperbolization ofpolyhedra, J. Differential Geom. 34 (1991), 347-388. [36] P. Deligne, Les immeubles de groupes de tresses generalises. Invent. Math. 17 (1972), 273-302. [37] W. Dicks and I.J. Leary, On subgroups of Coxeter groups. Geometry and Cohomology in Group Theory, P. KrophoUer, G. Niblo and R. Stohr, eds, London Math. Soc. Lecture Notes, Vol. 252, Cambridge Univ. Press, Cambridge (1998), 124-160. [38] A.N. Dranishnikov, The virtual cohomological dimension of Coxeter groups, Proc. Amer. Math. Soc. 125 (1997), 1885-1891. [39] R.D. Edwards, The topology of manifolds and cell-like maps, Proc. ICM Helsinki (1978), 111-127. [40] S. Eilenberg and T. Ganea, On the Lusternik-Schnirelman category of abstract groups, Ann. of Math. 65 (1957), 517-518. [41] R.H. Fox and L. Neuwirth, The braid groups. Math. Scand. 10 (1962), 119-126. [42] M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357^53. [43] R. Geroch, Positive sectional curvature does not imply positive Gauss-Bonnet integrand, Proc. Amer. Math. Soc. 54 (1976), 267-270. [44] E. Ghys and P. de la Harpe, eds, Sur les groupes hyperboliques d'apres Mikhael Gromov, Prog. Math., Vol. 83, Birkhauser, Boston (1990). [45] M. Gromov, Hyperbolic groups. Essays in Group Theory, M.S.R.I. Publ., Vol. 8, S.M. Gersten, ed.. Springer, New York (1987), 75-264. [46] A. Haefliger, Complexes of groups and oribhedra. Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger and A. Verjousky, eds, World Scientific, Singapore (1991), 504-540. [47] A. Haefliger, Extensions of complexes of groups, Ann. Inst. Fourier, Grenoble 42 (1-2) (1992), 275-311. [48] M. A. Kervaire, Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 (1969), 67-72. [49] F. Lanner, On complexes with transitive groups of automorphisms, Comm. Sem. Math. Univ. Lund 11 (1950), 1-71.

422

M.W.Davis

[50] H. van der Lek, The homotopy type of complex hyperplane complements, Ph.D. Thesis, University of Nijmegan (1983). [51] G. Moussong, Hyperbolic Coxeter groups, Ph.D. Thesis, The Ohio State University (1988). [52] C. Plant, Metric spaces of curvature > k, Handboolc of Geometric Topology, R.J. Davermann and R.B. Sher, eds, Elsevier, Amsterdam (2002), 819-898. [53] I. Rivin and C. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. I l l (1993), 77-111. [54] D. Rolfsen, Strongly convex metrics in cells. Bull. Amer. Math. Soc. 78 (1968), 171-175. [55] M. Ronan, Lectures on Buildings, Academic Press, San Diego (1989). [56] M. Salvetti, Topology of the complement of real hyperplanes in C", Invent. Math. 88 (1988), 603-618. [57] J.P Serre, Trees, Springer, New York (1989). [58] B. Spieler, Nonpositively curved orbihedra, Ph.D. Thesis, The Ohio State University (1992). [59] J.R. Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Phil. Soc. 58 (1962), 481-488. [60] J.R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88 (1968), 312-334. [61] R. Stanley, Flag f-vectors and the cd-index. Math. Z. 216 (1994), 483^99. [62] D. Stone, Geodesies inpiecewise linear manifolds. Trans. Amer. Math. Soc. 215 (1976), 1-44. [63] W. Thurston, The geometry and topology of 3-manifolds, Princeton University (1979) (reproduced lecture notes). [64] P. Thurston, The topology of 4-dimensional G-spaces and a study of 4-manifolds of non-positive curvature, Ph.D. Thesis, University of Tennessee (1993). [65] C. Tomei, The topology of the isospectral manifold of tridiagonal matrices, Duke Math. J. 51 (1984), 981996. [66] E.B. Vinberg, Discrete linear groups generated by reflections. Math. USSR Izvestija 5 (5) (1971), 10831119. [67] D.W. Walkup, The lower bound conjecture for 3- and 4-manifolds, Acta Math. 125 (1970), 75-107. [68] D. Wright, Contractible open manifolds which are not covering spaces. Topology 31 (1992), 281-291.

CHAPTER 9

Cohomological Dimension Theory Jerzy Dydak* University of Tennessee, Knoxville, TN 37996, USA E-mail: [email protected]

Contents 0. Historical development of ideas 0.1. Dimension 0.2. Cohomological dimension 1. Bockstein inequalities and Bockstein theorems 2. Cell-like maps 3. Edwards-Walsh complexes 4. Dranishnikov realization theorem 5. Infinite-dimensional compacta of finite integral dimension 6. Extension types and extension dimension 6.1. Classical cohomological dimension theory reinterpreted 6.2. Dual Bockstein algebra 7. Extension theory 8. Cohomological dimension and its connections to other branches of topology/mathematics 8.1. Compact group actions and Hilbert-Smith Conjecture 8.2. Gromov-Hausdorff metric and integral dimension 8.3. Cohomological dimension of groups 8.4. Mapping intersection problem 8.5. Rings of complex and real functions on compact metric spaces 8.6. Finitistic spaces 9. Open problems References

* Supported in part by grant DMS-9704372 from the National Science Foundation. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved 423

425 425 426 428 432 434 438 439 441 446 447 451 457 457 458 459 461 461 462 466 467

424

/ Dydak

Abstract The paper outlines the development of the cohomological dimension theory from its beginning (around 1930) to the present. The emphasis is on the main ideas and on the interaction between geometric intuition and the power of algebra. The historical development of ideas is mentioned with an explanation from the modem point of view.

Cohomological dimension theory

425

0. Historical development of ideas 0.1. Dimension One of the most intuitive geometric concepts is that of dimension. While it is very easy for a non-mathematician to guess the dimension of a particular geometric object, it is fairly difficult to define the dimension in a formal, rigorous way. The first mathematical definition of the dimension arises in linear algebra and one easily gets that the dimensions of the line, the plane, and the n-space (consisting of ordered n-tuples of real numbers) are respectively 1, 2, and n. After the formation of topology as an offshoot of analysis, mathematicians attempted to define dimension in a non-algebraic way. The most intuitive is the so-called small inductive dimension ind(X) of a space X. Essentially, ind(X) < n means that open sets U with ind(Fr U) ^n — 1 form a basis of X. 0.1. DEFINITION OF SMALL INDUCTIVE DIMENSION. Given a regular space X, one assigns to X the small inductive dimension ind X (also called the Menger-Urysohn dimension) which is an integer larger than or equal to —1, or infinity (oo), by the following conditions: (1) ind(X) = — 1 if and only if X is empty, (2) ind(X) < n, where 0 ^ n < oo, if for every point x e X and each neighborhood y C X of jc there is an open set (7 C V in X containing x so that ind(Fr L^) ^n — l, (3) ind(X) = n, where 0 < n < oo, if ind (X) ^ n and ind(X) ^n — I does not hold, (4) ind(X) = oo, if ind(X) < n does not hold for all /i < oo. Small inductive dimension ind works best for separable metric spaces. For normal spaces one has a traditional definition of the covering dimension dim. 0.2. DEFINITION OF COVERING DIMENSION. Given a normal space X, one assigns to X the covering dimension dim(X) (also called the Cech-Lebesgue dimension) which is an integer larger than or equal to —1, or infinity (oo), by the following conditions: (1) dim(X) ^ n, where 0 ^ n < oo, if every finite open cover of X has a finite open refinement of order at most n (this means that every n -\-2 distinct elements of the cover have empty intersection), (2) dim(X) = n, where 0 ^ n < oo, if dim(X) ^ n and dim(X) ^n — I does not hold, (3) dim(X) = oo, if dim(X) ^ n does not hold for all « < oo. There is a third way of defining dimension (the large inductive dimension) and we refer to [63] for a discussion of cases in which various topological dimensions coincide (it is so for the most significant class of topological spaces: the metrizable and separable spaces). We conclude this section by stating that in the case of Euclidean spaces the topological and algebraic dimensions are equal to each other. 0.3.

BROUWER'

S

THEOREM.

The algebraic and topological dimensions coincide for R^.

Historically, Theorem 0.3, also known as the Fundamental Theorem of Dimension Theory, is perhaps the first result relating algebraic/analytic and topological concepts. We refer

426

/ Dydak

the reader to [8] for a discussion of the ultimate theorem relating algebraic/analytic and topological ideas: the Atiyah-Singer Index Formula. 0.2. Cohomological dimension With the introduction of homology/cohomology groups it made sense to develop the concept of dimension in an algebraic fashion. The fundamental theorem is due to Alexandroff 0.4. ALEXANDROFF's THEOREM. IfX is a finite-dimensional compactum, then its dimension is the smallest integer n such that H^^^(X, A; Z) = 0 for all closed subsets A

ofX. One can replace Z by any abelian group G ^0 which leads to the notion of the cohomological dimension dimciX). 0.5. DEFINITION OF COHOMOLOGICAL DIMENSION. Given a paracompact space X and an abelian group G, one assigns to X the cohomological dimension dimG(X) which is an integer larger than or equal to —1, or infinity (oo), by the following conditions: (1) dimG(X) < n, where 0 ^ n < oo, if ^"+i(X, A; G) = 0 for all closed subsets A ofX, (2) dimG(X) = n, where 0 ^ n < oo, if dimG(X) ^ n and dimoiX) ^n — 1 does not hold, (3) dimG(X) = 00, if dimoiX) ^ n does not hold for all n < oo. The relevance of dimension with respect to arbitrary groups was underscored by Pontryagin [92] who realized that in order to construct compacta X and Y with dim(X x F) < dim(Z) H- dim(F), one has to construct a compactum P such that dim(P) > dimz/p(P) for some prime p. 0.6. PONTRYAGIN'S THEOREM. For each prime p there is a compactum Pp such that the following conditions hold: (1) dim(Pp) = 2, dimz/q(/*p) = 1 ifq is a prime and q^p, (2) dim(Pp X Pq) = 3 if q is a prime and p 7^ q. The year 1935 was crucial for the future development of the cohomological dimension theory. During the First International Topology Conference in Moscow, Alexandroff posed several questions (see [2]), and the following two turned out to be of significant importance. 0.7. ALEXANDROFF's PROBLEM. IS there a countable family Q of abelian groups such that for any compactum X and any abelian group G, the dimension dimG(X) can be expressed in terms of dim//(X), H eQl 0.8. ALEXANDROFF'S PROBLEM. IS there an infinite-dimensional compactum X such that dimG (X) < 00 for (a) G = Z, (b) s o m e G ^ O ?

Cohomological dimension theory

All

Problem 0.7 was solved by Bockstein [7] in 1956, who singled out the following groups (known now as Bockstein groups): 0.9.

DEHNITION OF BOCKSTEIN GROUPS.

The set Bg of Bockstein groups is

{Q}U U {Z/p,Z/p-,Z(p)}, p prime

where Z / p ^ is the p-torsion of Q/Z, and Z(p) consists of the rationals whose denominator is not divisible by p. In the process of solving Problem 0.7, Bockstein discovered his exact sequence which led to cohomology operations, an important concept in algebraic topology (see [103]). Pontryagin made a serious attempt to solve Problem 0.8. He knew that spheres are closely related to the concept of dimension (see 0.12) and felt that first one needs to get a better understanding of the homotopy groups of spheres. This led to another sizable part of algebraic topology. Thus, cohomological dimension theory is, historically, part of the bedrock of modem algebraic topology. The first part of the development of the cohomological dimension theory culminates with the publishing of the survey article by Kuzminov [80], which is a 'must read' even today. The second part of the development of the cohomological dimension theory was jump started by the extremely elegant paper of Walsh [107]. Generally speaking, [80] relies on algebra as the driving force, while in [107] it is the geometry which plays the dominant role. In the view of this author, the following aspects of [107] were decisive in awarding it the role of a milestone: (a) it established a bridge between geometric topology and the cohomological dimension theory in the form of cell-like maps, (b) it pursued, for the first time, a formal geometric analogy between covering dimension and the integral cohomological dimension, (c) it introduced a geometric construction, known as the Edwards-Walsh complex, which turned out to be a very useful tool. The Edwards-Walsh complex provided the geometry in Dranishnikov's [28] solution of Problem 0.8. The algebra consisted of computing the A'-theory groups of the EilenbergMacLane complex K{TJ, 3) (details later on). By analyzing the cohomology exact sequence of a pair (X, A) > ^ " ( X ; G) -^ H'^iA; G) -^ 7f"+^(X, A; G) ^ - it was known that dimciX) ^ n is equivalent to K(G, n) being an absolute extensor of X (see 0.11). First, let us recall the definition of absolute extensors. 0.10. DEFINITION OF ABSOLUTE EXTENSOR. K is m absolute extensor of X (notations: K e AE(X) or XTK) if every map f .A^- K extends over Z if A is closed in X. 0.11. C O H E N ' S K(G,n)eAE(X).

THEOREM.

If X is locally compact, then dimoiX) ^ n if and only if

428

/ Dydak

In the case of the covering dimension one has a similar result. 0.12. ALEXANDROFF'S THEOREM. IfX is compact, then dim(X) ^ n if and only if S"" e AE(X). The strength of [107] is that statements 0.11-0.12 were chosen as definitions rather than theorems, which resulted in elegance and simplicity. This idea was pursued by Dranishnikov [32], Dydak [47] and Dranishnikov and Dydak [38]. As a result, we are currently at the third stage of the development of the cohomological dimension theory, known as Extension Theory. We refer the reader to the classic work [68] for an account of the dimension theory of separable metric spaces. The starting point of that book is the small inductive dimension. It contains a proof of 0.3 on p. 41 (for the original proof see [16]) and a proof of 0.12 on p. 83. For the most up to date account of dimension theory, including the relations between various concepts of dimension, we refer to [63]. Since Alexandroff is the founder of cohomological dimension theory, it is not surprising that most early research in that area was done in the former Soviet Union. We recommend papers by Bockstein, Boltyanskij, Kuzminov, Pontryagin, Shvedov, Sitnikov, Sklyarenko, and Zarelua (see our list of references) if one wants to get a taste of the achievements of the Soviet School of Topology. Outside of the former Soviet Union, there was also research done by Borsuk, Cohen, Dyer, Kodama, and Nagata. As of now, there is no book which covers a sizable part of cohomological dimension theory. There is a set of unpublished notes [40], and [85] contains a chapter devoted to cohomological dimension theory (one can find a proof of 0.4 there). Bredon's book [14] contains some results for cohomological dimension with coefficients in principal ideal domains. We recommend [80] for a discussion of the pre-Dranishnikov era (one can find a proof of 0.11 there), and we recommend [27] for an account of Dranishnikov's solutions of all the major problems of classical cohomological dimension theory. Chigogidze [18] has extended several results of cohomological dimension theory of metrizable spaces to Tychonoff spaces.

1. Bockstein inequalities and Bockstein theorems To tackle Problem 0.7, Bockstein needed a way to compare the cohomological dimensions of a space with respect to various groups. This was achieved via his exact sequence.

1.1.

BOCKSTEIN's EXACT SEQUENCE.

closed subspace. If 0 ^^ Gi -^ Gj^ then there is an exact sequence

Suppose X is a topological space and A is its

G3 -^ Ois a short exact sequence ofabelian groups,

> H'^iX, A; GO -^ ^ " ( X , A; G2) -^ H"(X, A; G3) -> //"^^^ (X, A; Gi) ^ • • •. Bockstein's exact sequence immediately implies a set of inequahties.

Cohomological dimension theory

429

1.2. GENERAL BOCKSTEIN'S INEQUALITIES. Suppose X is a topological space and 0 ^ G{ -^ G2 ^ G3 ^ 0 is a short exact sequence ofabelian groups. Then the following inequalities hold: (a) dimG2 (^) ^ max(dimGi {X), dimG3 {X)), (b) dimGi(X) < max(dimG2(^), dimG3(X) + 1), (c) diniGgOT ^ max(dimGi(X) — 1, dimG2 (^))In the case of Bockstein groups, general Bockstein inequalities can be reduced as follows: 1.3. BOCKSTEIN'S INEQUALITIES. Suppose X is a topological space. Then the following inequalities hold: (1) dimz/poo(X) ^ dimz/p(X), (2) dimz/p(X) ^ dimz/poo(Z) + 1, (3) dimQ(Z)
FIRST BOCKSTEIN'S THEOREM.

IfX is compact, then

dimG(X) = max{dim//(X) \H

ecf{G)].

The set (7(G) appearing in the theorem above is known as the Bockstein basis of G, and is defined as follows: 1.5. DEFINITION OF THE BOCKSTEIN BASIS. Given an abelian group G its Bockstein basis a{G) is the subset of Eg defined as follows: (1) Q G aCG) if and only if Q 0 G 7^ 0, (2) Z/p G a{G) if and only if (Z/p) 0 G / 0, (3) Z(p) G or(G) if and only if ( Z / p ^ ) O G 7^ 0, (4) Z / p ^ G a(G) if and only if ( Z / p ^ ) * G 7^ 0 (here / / * G is the torsion product of groups H and G). Since Pontryagin's example (see Theorem 0.6) demonstrated that the logarithmic law dim(X X 7) = dim(X) + dim(y), known to be true in the case of Euclidean spaces, does not hold for compacta, it was natural to seek formulae describing dimG(X x Y) in terms of the cohomological dimensions of X and Y. This problem was completely solved by Bockstein as follows: 1.6. SECOND BOCKSTEIN THEOREM. Suppose X and Y are compact. Then, (1) dimz/p(X X 7) =dimz/p(X) +dimz/p(F), (2) dimQ(X X r ) = dimQ(Z) + dimQ(F),

430

/ Dydak

(3) dimz/pcx) (X X 7) = max{dimz/p^ (X)+dimz/poo (7), dimz/p(X)+dimz/p(F) - 1 } , (4) dimZ(p)(X X 7) = dimz^p^CX) + dimz^p^CF) if dimz^^^{X) = dimz/p-(X) or dimz(p)(7) = dimz/poo(y), and (5) dimz(p) iXxY)= max{dimz/pc- (X x F) + 1 , dimQ(X) + dimQ(y)} if dimz^^^ (X) > dimz/poo(X) anJ dimz(p) (F) > dimz/poo(y). Here is the geometric reason why the cohomological dimension works better in the case of Cartesian products of compacta: K e AE(X x F) is a stronger condition than K^ e AE(X) (here, A'^ is the space of maps from F to ^ in the compact-open topology). Indeed, any map f: A -> K^, A being closed in X, induces a map f^:AxY^^K via f\a, y) = f{a){y). If f extends over X x F, then / extends over X. If K is an Eilenberg-MacLane space K{G, n), then A'^ is known to be the product of EilenbergMacLane spaces K(H^~^ (F; G), /), 0 ^ / ^ n (this is a geometric version of the Kunneth Formula, for a more general result see [26]). On the other hand, if A^ is a sphere (recall that the covering dimension is related to maps into spheres), the space K^ is much more complicated. The First Bockstein Theorem means that the cohomological dimension of a compactum X is determined by the set {dim// iX)}HeBg» ^^^ the Bockstein Inequalities impose certain restrictions on this set. This leads to the notion of a Bockstein function. 1.7. DEFINITION OF BOCKSTEIN FUNCTION. A Bockstein function is a function D : Bg ^^ Z^o from the set of Bockstein groups to the set of non-negative integers plus infinity such that the following holds: (1) Z)(Z/p^) < Z)(Z/p), (2) D ( Z / p ) ^ D ( Z / p ^ ) + l, (3) D(Q) < Z)(Z(p)), (4) D(Z/p) ^ D(Z(p)), (5) Z ) ( Z / p ^ ) < m a x ( D ( Q ) , D ( Z ( p ) ) - l ) , (6) D(Z(p))^max(D(Q),D(Z/p^) + l), (7) ifminD = 0,thenD = 0. Obviously, the basic example of a Bockstein function is the one induced by a compactum. 1.8. DEFINITION OF DIMENSION FUNCTION OF A COMPACTUM. Given a compactum X, its dimension function Dx is the Bockstein function defined by Dx(H) = dim//(X). A natural question is: 1.9. BOCKSTEIN-BOLTYANSKI PROBLEM. Is every Bockstein function equal to the dimension function of a compactum? This problem was completely solved by Dranishnikov (details later on). Notice that, in view of the First Bockstein Theorem 1.4 and the Alexandroff Theorem 0.4, the dimension of a compactum corresponds to the well-known concept of the supremum of a function:

Cohomological dimension theory

1.10. DEHNITION OF THE DIMENSION OF A BOCKSTEIN dimZ) of a Bockstein function is sup{D(H) \ H eBg}.

431 FUNCTION.

The dimension

It turns out one can translate operations on compact spaces into operations on Bockstein functions. 1.11. DEFINITION OF A WEDGE OF BOCKSTEIN FUNCTIONS. Given a set {Da}aeA of Bockstein functions, one defines its wedge VaeA ^« ^y y Da(H) =

max{Da(H)\aeA}.

aeA

1.12. DEFINITION OF THE PRODUCT OF BOCKSTEIN FUNCTIONS. Given two Bockstein functions Di and D2, one defines their product D\ x D2 as follows: (1) (Di X D2)(Z/p) = Di(Z/p) + D2(Z/p), (2) (Di X D2)(Q) = Di(Q) + D2(Q), (3) (Di x D 2 ) ( Z / p ^ ) = m a x { D i ( Z / p ^ ) + Z)2(Z/p^),Di(Z/p) + Z ) 2 ( Z / p ) - l } , (4) (Di X D2)(Z(p)) = Z)i(Z(p)) + Z)2(Z(p)) if Di(Z(p)) = D i ( Z / p ^ ) or D2(Z(p)) = D2(Z/p~),and (5) (Di X D2)(Z(p)) = max{(Z)i x D2)(Z/p^) + 1, Di(Q) + D2(Q)} if Di(Z(p)) > D i ( Z / p ^ ) and D2(Z(p)) > D2(Z/p^). In this way one gets an algebra: 1.13. DEFINITION OF BOCKSTEIN ALGEBRA. The Bockstein algebra {BT, v, x) is the set of all Bockstein functions B!F equipped with two associative binary operations v and X so that V is distributive with respect to x. Now, the Second Bockstein Theorem 1.6 can be reformulated as: 1.14. THEOREM. IfX and Y are compact, then (1) DxwY = DxV DY, (2) DxxY = Dxy Dx of the algebra of pointed compact spaces with operations of wedge and smash product to the Bockstein algebra. The significance of Theorem 1.15 is that, quite often, one does calculations in the Bockstein Algebra and interprets them in terms of geometrical properties of X. The best source for all the results of this section is [80]. Notice that 1.5 is taken from [40] and is equivalent to the definition of the Bockstein basis which can be found in [80].

432

/ Dydak

2. Cell-like maps The connection between cohomological dimension and geometric topology in the form of cell-like mappings was established in [107]. 2.1. DEFINITION OF CELL-LIKE SETS/MAPS. A space A is cell-like if any map from A to a CW complex is nuU-homotopic. The map / : X -> 7 is cell-like if / is proper and f~^(y) is cell-like for all y € 7. Cell-like sets and maps are more suitable in point set topology, while in geometric topology one has the related concept of cellular subsets/maps. 2.2. DEFINITION OF CELLULAR SUBSETS/MAPS. A compact subset A of an n-manifold M" is cellular if open n-cells containing A form a basis of neighborhoods of A. The map / : M^ -^ 7 is cellular if / is proper and f~^{y) is a cellular subset of M" for all j G 7. Cellular maps arise in almost every question involving the existence of homeomorphisms between manifolds. The first significant application of the concept of cellularity was demonstrated by M. Brown in his proof of the Generalized Schoenflies Conjecture. The most basic result is that cellular maps between closed manifolds are nearhomeomorphisms. 2.3. THEOREM. Suppose M" and N^ are closed n-manifolds. The closure of the set of homeomorphisms from M" to N'^ in the compact-open topology coincides with the set of all cellular maps from M^ to N^. In practice, instead of a cell-like map between two manifolds, one has a cell-like decomposition G of a manifold M" and one wonders if the natural projection n : M" -> M^ /G is a near-homeomorphism. The fundamental result in this area is:

2.4. CELL-LIKE APPROXIMATION

THEOREM.

Suppose G is a cell-like decomposition of

an n-manifold M", where n^ 5. Then n : M" -> M^ / G is a near-homeomorphism if and only ifM^/G is finite-dimensional and has the disjoint disks property. Thus, one arrives at the following problem. 2.5. CELL-LIKE MAPPING PROBLEM. If / : X ^ dimensional compactum, is 7 finite dimensional?

7 is cell-like and X is finite-

This is the point of overlap of geometric topology and cohomological dimension theory. Namely, [107] contains a characterization of cell-like images of n-dimensional compacta. 2.6. EDWARDS-WALSH THEOREM. The compactum Y is a cell-like image of an n-dimensional compactum if and only if the integral cohomological dimension ofY is at mostn.

Cohomological dimension theory

433

In this way one gets the equivalence of the Cell-Hke Mapping Problem 2.5 and the Alexandroff Problem 0.8(a). In point set topology one has a generic construction of cell-like maps (see [69]). We will present this construction in the more general setting of shape theory. 2.7. DEFINITION OF A SHAPE EQUIVALENCE. A map / : X ^ F is called a shape equiv-

alence if, for any CW complex K, the induced function / * : [F, K] -^ [X, K] of sets of homotopy classes is a bijection. / is called a hereditary shape equivalence if, for any closed subset A of F, the map f\f~^{A): f~^{A) ^- A is a shape equivalence. Notice that X is cell-like if and only if the unique map from X to any one-point space is a shape equivalence. In particular, each hereditary shape equivalence is a cell-like map. We are ready to present the general construction of cell-like maps. 2.8. THEOREM. For every compactum X there is a hereditary shape equivalence n :X ^^ X such that X is the inverse limit of finite simplicial complexes with bonding maps being simplicial. PROOF. TO a shape

theorist, the idea of the construction is quite simple: any compactum is shape equivalent to an inverse sequence {JT^^^ : X „ + I ^^ X „ } of finite simplicial complexes with bonding maps being simplicial (see [55] or [82]). The inverse limit X of {n^'^^: Xfi+i -^ Xfi} is also shape equivalent to X (see [55] or [82]). Therefore, one would expect X and X to be shape equivalent via a map. Let us show this in a more detailed manner. For each n ^ I choose a finite open cover Un of X so that the diameter of each element of Un is less than 2~^ and ZY„+i refines ZY„. Let Xn be the nerve of Un and let 7r^+^: Xn+i ^- X„ be a simpHcial map determined by choosing, for each U e Un^\, an element U' G tin containing U. Let X be the inverse limit of [n^^^: Z„+i -> Xn). Given a point z'mXn, z can be expressed as a linear combination ^i=\Ci • Ui, where Ui € Un for / ^ k, Ci > 0 for / ^ k, Y^i^x Ci = 1, and n?=i ^i T^ 0- ^^ define the carrier carr(z) of z to be the closure of H/^i ^i • Notice that the diameter dmm(carr(z)) of carr(z) is less than 2~" and carr{z) C carr{7t^_^{z)). Thus, if x = {Zn}n^\ ^ ^ . then P l ^ i carr(zn) consists of precisely one point, denoted by 7r(jc). It is easy to check that TT : Z ^ - Z is continuous, so it suffices to show that TT is a shape equivalence. (Indeed, TT |7r~^ (A) :7t~^(A) -> A can be obtained from A in the same manner as n was obtained from X. Thus, n\7t~^ (A) is a shape equivalence for each closed subset A of X, which means that TT is a hereditary shape equivalence.) For each n we have the natural projection Ttn'-X ^^ Z„, and there IS a map pn '. X —> X^ induced by a partition of unity associated with Un. This means that if pn (x) = 5Z/^i c/ • Ut, where Ui € Un for / ^ k, ci > 0 for / ^ k, and XlLi ^i = 1' ^^^^ ^ ^ PlLi ^i- ^^^h {^n}n^i and {pn}n^\ are shape equivalences (see [55] or [82] on how to interpret this statement). Thus, to show that TT is a shape equivalence, it suffices to prove that /?„ o TT is homotopic to itn- Suppose z = {zj}j^\ G X and Zn = Xl/=i ^i ' ^i^ where Ui e Un for / ^ k, Ci > 0 for / ^ k, and J2i=i ^i = 1- We know that 7r(z) G cKplf^j Ui). Suppose pn(7t(z)) =

434

/ Dydak

Yll=\ ^s 'Vs, where ds > 0 for each s. In view of the way pn was described, we know that Jt(z) e n L i ys. Consequently, ( f l t i ^i) ^ (HJ^i Vs) / 0, and Vi,..., V,, ( / i , . . . , ^^ form a simplex of X„. Now, the formula H(z, t) = t - 7t(z) -\- (I — t) - pn(7t(z)) gives a homotopy joining pnon and n. D Notice that if one defines X^^^ to be the inverse limit of the maps in the ^-skeleta, K^^ • ^f+i -^ ^^n\ then X^o^ is a Cantor set (unless X is discrete), and n: X^^) -> X is onto. Thus, one recovers the well-known fact that every compactum is an image of a Cantor set. More generally, TT : X^^^ -> X is a L^y^~^-map (see [74] for a definition of C/y^-maps). Thus, every compactum is an image of a /^-dimensional compactum via a ^y^-i-map. We refer the reader to [83] for the history of the Cell-like Mapping Problem and Theorem 2.3. A proof of Theorem 2.4 can be found in [21]. See [78,79], and [108] for cases in which the Cell-like Mapping Problem has a positive solution. The Edwards-Walsh Theorem 2.6 has been generalized to non-compact metric spaces by Rubin and Schapiro [96]. Other generalizations involve cohomological dimension with groups different from integers; e.g., Dranishnikov [27] considered integers modulo p and compact metric spaces, where the property of a map being cell-like is replaced by the property of being acyclic. The case of non-compact metric spaces was considered by Koyama and Yokoi [76]. A different generalization of the Edwards-Walsh Theorem appears in [74] (the compact metric case) and [75] (the non-compact metric case). Namely, the authors characterize spaces which are images of UV^~^ - and acyclic maps (with respect to a ring with unity) of n-dimensional spaces.

3. Edwards-Walsh complexes As we mentioned (see 0.12), the covering dimension of compacta can be characterized in terms of extension of maps into spheres. There is a dual characterization in terms of approximate lifting of maps. It is well-known that X is at most n-dimensional if and only if any map from X to a CW complex L can be approximated by a map to the n-skeleton L^"^ of L. This property of «-dimensional compacta can be restated in terms of approximate liftings. First, we need to introduce the concept of a combinatorial map. 3.1. DEFINITION OF COMBINATORIAL MAPS.

A map TT : L ^

L of CW complexes is

called combinatorial provided TT"^ (^) is a subcomplex of L whenever A' is a subcomplex ofL. 3.2. DEFINITION OF APPROXIMATE LIFTS. Suppose TT : L ^ L is a combinatorial map

and L is a simplicial complex with the CW topology. A map f :X ^^ L is said to approximately lift with respect to n provided there is a map f :X ^^ L such that, for each x e X and each simplex Ain L, f(x) e A implies n o f(x) e A. Now, we can restate the property of approximating maps by maps into the n-skeleton in terms of approximate lifts.

Cohomological dimension theory

435

3.3. THEOREM. X is at most n-dimensional if and only if, for any simplicial complex L, every map f :X ^^ L has an approximate lift with respect to the inclusion L^^^ ^^ L. One of the goals of [107] was to generalize this result to the case of the integral cohomological dimension. This result is mentioned in the abstract of [60] but a proof was never published. 3.4. THEOREM. Let n be an integer. For each simplicial complex L there is a CWcomplex EWziL, n) and a combinatorial map TTL : EWz{L, n) ^^ L so that the following conditions are equivalent: (a) dimz(X) < n, (b) for any simplicial complex L, every map f :X -^ L has an approximate lift with respect to TTL. Before outlining the details of the construction of the Edwards-Walsh complexes EWz{L, n), let us discuss a simpler construction which stems from the work of Pontryagin and which can be viewed as a precursor of the Edwards-Walsh construction. To simplify our exposition we will deal with ordered simplicial complexes, i.e., complexes with all their vertices being ordered (for example, by enumeration). Ordered simplicial complexes form a category, with morphisms being simplicial inclusions which preserve the order of vertices. We will consider its subcategory SC2 consisting of ordered simplicial complexes of dimension at most 2. 3.5. DEFINITION. SC (SC2) is the category of ordered simplicial complexes (of dimension at most 2). Morphisms of SC (SC2) are simplicial inclusions which preserve the order of vertices. CW (CW2) is the category of of CW complexes (of dimension at most 2) with all cells being attached via PL-maps (in this way one can give a PL-structure to any of their finite subcomplexes). Morphisms of CW (CW2) are continuous maps which induce an isomorphism of the domain with the image of the map. 3.6. THEOREM. There is a functor Pz/pfrom SC2 to CW2 and there is a natural transformation TCL • Pz/p(L) -^ L of functors with the following properties: (a) Suppose A is a closed subset of an at most 2-dimensional compactum X so that dimz/p(X) < 1. Given an ordered simplicial complex L with dim(L) < 2, given a map g:A^^ Pz/p(L), and given an extension f :X -^ L of 7ii o g, there is an extension G:X -^ Pz/p(L) of g so that G is an approximate lift of f with respect tOTtL.

(b) Suppose X is an at most 2-dimensional compactum. If any map f :X ^^ Lfrom X to a 2-dimensional ordered simplicial complex L has an approximate lift with respect to 7TL, then dimz/p(X) < 1. (c) Given an object L of SC2, given a subcomplex K of L, and given a map f: K -^ K(Z/p, 1), the map f OTTK extends over Pz/p(L). PROOF. If L is of dimension at most 1, then we put Pz/piL) = L and TTL : Pz/p(L) -^ L is the identity map. Now, it suffices to define Pz/p{A) for the standard 2-simplex A

436

/ Dydak

with vertices fo, v\, V2 so that VQ < v\ < V2. We attach a 2-cell A^ via a PL-map a ^ ' -> dA of degree /?. Thus, A' is the 2-skeleton of a K(Z/p, 1), with az\ being its 1-skeleton. Since Z\ is contractible, dA^ ^^ dA extends to TTL : A^ ^^ A, giving us the map ^L • Pz/p(^) = A! -^ A. Suppose L is an ordered simpUcial complex of dimension at most 2. Given a 2-simplex 5 of L there is a unique isomorphism is'.A^^s. Pi^i^iL) is the union of adjunction spaces A! U/^|a^ L^^^ and 7ii: Pz/^{L) -^ Lh the unique map so that 7TL\A' ^is\dA L^^^ is the identity on L^^^ and equals is o TTA on A^ C ^^ '-'/5|az\ ^^^^• Notice that Pz/p(A), being the 2-skeleton of of a K(Z/p, 1), is an absolute extensor of any, at most 2-dimensional, compactum X with dimz/pCA') ^ 1. Indeed, given a closed subset A of X, and given / : A -> Pz/p(A), f can be extended to F:X ^^ ^ ( Z / p , 1). Since dim(X) ^ 2, F can be approximated, rel. A, by F ' : X -> Pz/p(^) = ^ ( Z / p , 1)^^\ Now, condition (a), in the case L = A, follows from the fact that Pz/^{A) is an absolute extensor of X. Since any Pz/^{L) is built by putting copies of Pz/^{A) together, condition (a) follows. Implication (b) is weaker than implication (c). Before showing it, let us make sure implication (c) holds. As before, it suffices to show it in the case of L being the standard 2-simplex A. Suppose ^ is a subcomplex of A and f :K ^^ ^ ( Z / p , 1) is a map. Notice that / can be extended over the 1-skeleton of A. Thus, without loss of generality, it suffices to consider the case K = dA. Since the attaching map a'.dA'^^dA'isoi degree p,ao f is null-homotopic and extends over A'. Now, let us show that impUcation (c) impUes implication (b). Suppose X is an at most 2-dimensional compactum, A is a closed subset of X, and g'.A^^ ^ ( Z / p , 1) is a map. There is an at most 2-dimensional simplicial complex L (L can be obtained as the nerve of a finite open cover of X of order at most 2), there is a map n :X -^ L, there is a subcomplex K of L, and there is a map g' :K ^^ KiTj/^j^, 1) so that 7r(A) C K and g' o (TT I A) is homotopic to g. Choose an approximate lift 77 : X -^ Pz/p(L) of 7t with respect to TtL. Also, choose an extension G: Pz/p(L) -> K(Z/p, I) of g^ OTTK- Notice that G o H is a homotopy extension of g. Thus, dimz/p(X) ^ 1. D The Edwards-Walsh complex EWz(L, n) is constructed by generalizing the procedure of Pontryagin. For simplicity, we fix n and denote EWz(L, n) by EWzi^). Fii'st of all, the «-skeleton ofEWziL) is equal to the w-skeleton of L, and TTL is the identity map on the n-skeleton of EWz{E). Given an {n + l)-simplex Z\ of L one creates EWzi^^) as a K{Z, n) complex with dA being its {n + l)-skeleton. 3.7. THEOREM . Suppose n is an integer. There is a functor EWz from SC toCW and there is a natural transformation TTL \EWZ{E) -> L of functors with the following properties: (a) Suppose A is a closed subset of a compactum X so that dimz(X) ^ n. Given an ordered simplicial complex L, given a map g'.A^^ EWz(L), and given an extension f:X->Lof7tLog, there is an extension G :X ^- EWz(L) of g so that G is an approximate lift of f with respect to TVL(b) Suppose X is a compactum. If any map / : X ^- Lfrom X to an ordered simplicial complex L has an approximate lift with respect to TTI, then dimz(X) ^ n. (c) Given an object L of SC, given a subcomplex K of L, and given a map f '.K ^^ K{7J, n), the map f OTTK extends over EWz{E).

Cohomological dimension theory

437

(d) The (n + I)-skeleton of EWz(L) coincides with the n-skeleton of EWz(L) and TtilEWziL)^''^ :EWz(L)^''^ -^ L^""^ is an isomorphism of CWcomplexes. SKETCH OF PROOF. lfn<2, take EWziL) = L^"^ and let JTL be the inclusion. The reason this works is that S^ is a K(Z, 1). Thus, assume n ^ 2. If L is of dimension at most n, then we put EWz(L) = L and let TTL '- EWz(L) -> Lbc the identity map. Now, it suffices to define EWz(^k) for all standard simplices Ak with vertices vo,v\,V2,.--yVk so that VQ < v\ < V2 < " ' < Vk and / : ^ n + l.If/: = n + l,we attach cells to the n-skeleton of An+\ in order to get EWziAn+x) as an Eilenberg-MacLane complex K{7J, n). Obviously, we can do that without attaching cells in dimension n -\- \. Since An^\ is contractible, the inclusion dAn+\ -^ ^«+i extends to TTL :EWz(An-\-\) -^ ^n+i- Once EWz{An^\) is known for all ordered simplices of dimension at most m, one can define EWz(L) for all ordered simplicial complexes of dimension at most m. Indeed, suppose L is an ordered simpHcial complex of dimension at most m. Notice that EWz(L) must contain a copy of EWz(L^^~^^) as a subcomplex. So it makes sense to build EWz(L) by attaching cells to EWz(L^'^~^'>), Given an m-simplex s of L, the inclusion is: ds -^ L^^ '^ induces an inclusion js'.EWzids) -> EWz{L^'^~^'>). EWz{L) is the union of adjunction spaces EWziE^^'^"^) Uy^ EWz{s), and TTL :EWZ(L) -^ L is the unique extension of n^im-i) : £ W Z ( L ( ^ - ^ ^ ) -^ L^^-^^ so that jtilEWzis) equals its for all s. Thus, EWz(L) is known for all ordered simplicial complexes of dimension at most n + 1. Suppose that m > n, EWz(L) is known for all ordered simpUcial complexes of dimension at most m, and EWz(Ak) is a K(Z^^^\n) for each n < k ^m. Notice that EWz(dAm-{-\) is simply connected with the same integral homology (in dimensions from 1 to n) as A^_^^. As if„(zlj^^j) is free, it is isomorphic to z^^^+^^ for some integer a(m +1). Thus, 7tn(EWz(^Afn+i)) is isomorphic to z^^^+^^ and one can attach cells of dimension at least « + 2 to EWz(dAm+i) in order to create EWziAm+i) as a K(Z''^'^^^\ n). This completes the inductive process of defining EWz{E) for all ordered simplicial complexes L. As in 3.6 one shows condition (a), and one shows that (b) is weaker than (c). Thus, it suffices to prove that (c) is weaker than (d) (the validity of (d) follows from our construction). As in 3.6, it suffices to show this for the case of L being the standard simplex AmSuppose ^ is a subcomplex of Am and / : ^ ^- ^ ( Z , m) is a map. Notice that / can be extended over the ^z-skeleton of Am- Thus, without loss of generality, it suffices to consider the case K D A^- Since there are no {n + l)-cells in EWz{L) — EWz^K), any map EWziK) -^ K{Z, n) extends over EWz{L). D THE IDEA OF THE PROOF OF EDWARDS-WALSH THEOREM

2.6 {Sketch). Suppose X

is of integral cohomological dimension n, where n > 1, and we would like to construct a cell-like map f '.X' ^^ X with X' being of dimension n. A natural guess is to modify 7t: X^^~^^^ -> X (see 2.8 and the comment after its proof). The reason is that TT is a canonical map which is almost cell-like, namely, it is a t/V"-map. However, Z^"+^^ is, in general, (n + 1)-dimensional. The idea is to replace it by an n-dimensional compactum. Consider the projection X -^ Xk, Xk being the nerve of an open cover of X of mesh at most 2~^. Since X is of integral cohomological dimension n, there is an approximate lift X -^ EWziXk, n) which factors as X -> X^ ^- EWziXk, n) for large m. As you recall.

438

/ Dydak

the (n + l)-skeleton ofEWz(Xk, n) coincides with its M-skeleton. Thus, X^Jil^^^ -^ X^^^^'^ can be approximately factored (or replaced by a map) through an n-dimensional CW complex. This is a rough idea of how to construct a cell-like map. The details are, obviously, more complicated. D Because of the usefulness of the Edwards-Walsh complexes EWz{L, /i), one may ponder their existence for groups G other that the integers Z. One may extract the essential features ofEWz as follows. 3.8. DEFINITION OF EDWARDS-WALSH FUNCTOR EWQ- Suppose n is an integer.

A functor EWQ : <SC ^- C W is an Edwards-Walsh functor provided there is a natural transformation of functors TTL : EWQ (L) -^ L such that the following conditions are satisfied: (a) TtL is a combinatorial map and 7TJ^\K) = EWG (K) for every subcomplex K of L, (b) EWG(^) is a K(G^,n) for each ordered simplex A (here m depends on A), (c) EWG(L) = L and TTL is the identity if dim(L) ^ n, and (d) if A^ is a subcomplex of L and f :K ^^ K(G,n) is a map, then f OTTK extends over EWG (L). The following is the most general result on the existence of Edwards-Walsh functors. 3.9. THEOREM. An Edwards-Walsh functor exists for each abelian group G such that there is a homomorphism of: Z -> G satisfying the following properties: (a) a<S>id:Z(^G ^^ G (S>G is an isomorphism, and (b) Of* :Hom(G, G) -^ Hom{Z, G) is an isomorphism. Theorem 3.9 implies the existence of Edwards-Walsh functors for all Bockstein groups but Z / p ^ . In the latter case, an Edwards-Walsh functor fails to exist. Edwards-Walsh complexes were introduced in [107]. Theorem 3.9 was claimed in [60] under the weaker assumption that a* is an epimorphism (instead of being an isomorphism). Koyama and Yokoi [77] detected a gap in [60] and provided details that of* being an isomorphism fills the gap.

4. DranishnikoY realization theorem Dranishnikov realized the potential of Edwards-Walsh complexes in solving Problems 0.8(b) and 1.9. He used related constructions called modifications. We present an outline of Dranishnikov's work using Edwards-Walsh complexes as in [60]. First, let us recall Pontryagin's construction of a 2-dimensional continuum X with dimz/2 X =1 (compare with Theorem 0.6). 4.1. EXAMPLE OF X WITH dimz/2(^) = 1, dim(X) = 2. Construction. X is defined as the inverse limit of the sequence • • • X^+i -^ X^ -^ • • •, where Xi = 5^ is the 2-sphere, Xk+\ = Pz/iiXk) and Xk-\-\ -> Xk is equal to TTX^, with Xk being considered with a triangulation with simplices being sufficiently small in a given metric on Xk.

Cohomological dimension theory

439

Since TZL • ^z/2(^) -^ L induces an isomorphism of rational homology, X is of covering dimension 2. It remains to show that dimz/2(^) = 1- This property is weaker than the analogous property of the map Tti: Pz/2(L) -> L: given a subcomplex A' of L and given f \K ^^ K{TJ/2, 1), the composition f OTTK extends over Pz/2(^)Dranishnikov solved Problem 0.8(b) positively by constructing compacta A„, « > 3, so that dimz/2(^«) = 1, dim(A„) = n. The one-point compactification A of the discrete sum of all An,n^3, satisfies dimz/2(^) = 1 and dim(A) = oo. We will outline the construction in the case of n = 5. 4.2. EXAMPLE OF X WITH dimz/2(^) = 1, dim(X) = 5. Construction. A natural attempt is to define X as the inverse limit of the sequence • • • Xy^+i ^- X^t ^^ • • •, where X\ = S^ is the 5-sphere, X^^+i = EWz/2i^k, 1) and Xk-\-\ -> Xk is equal to nxi,- The major difference between the current construction and Example 4.1 is that EWz/2(Xk, 1) is not a finite CW complex. Therefore, it would be impossible to achieve the compactness of X. However, we may try to pick Xk-\-\ as a finite subcomplex of EWz/ii^k, !)• This would ensure the compactness of X and, as in Example 4.1, X would be of Z/2-cohomological dimension 1. The only remaining task is to make sure that X is of dimension 5. This is done by keeping H^(X, Q) non-zero as follows. The map TTL .EWz/iiL, 1) -> L induces isomorphisms of all cohomology groups with rational coefficients (this is a combinatorial version of the Vietoris-Begle Theorem - notice that point-inverses of simplices have trivial cohomology groups with rational coefficients). If c e H^{L; Q) is non-zero, we choose CL G H^(EWZ/2(L, 1); Q) such that CL = (:TL)*(C), and we use the fact that the cohomology theory with rational coefficients is continuous, i.e., there is a finite subcomplex K of EWz/2(L, 1) with CL restricted to K being non-zero. Obviously, we may replace K by its 5-skeleton. Now, use this operation inductively starting with c\ e H^(S^; Q) being a generator. By refining the technique of Example 4.2, Dranishnikov was able to solve Problem 1.9 of Bockstein-Boltyanski in positive.

4.3. DRANISHNIKOV's REALIZATION THEOREM. For every Bockstein function D, there is a compactum X with Dx = D and dim(Z) = dim(D). One can find the original proof of Theorem 4.3 in [27]. This proof was later modified in [60] by an extensive use of Edwards-Walsh complexes.

5. Infinite-dimensional compacta of finite integral dimension The next stage in Dranishnikov's program was to solve Problem 0.8(a). It makes sense to mimic Construction 4.2 and attempt to define an infinite-dimensional compactum X of finite integral cohomological dimension as the inverse Umit of the sequence • • • Z^+i ^• Xk ^^ '", where X\ = S^ is the n-sphere, Xk+\ C EWz{Xk,m) and Xk^\ -^ Xk is equal to TTxJXy^+i. Since K(Z,n) has non-trivial cohomology with rational coefficients, one

440

/ Dydak

needs to find another continuous cohomology theory which vanishes on K(Z, n). Luckily, such a cohomology theory already existed in the literature. Namely, the reduced complex ^-theory with mod p coefficients vanishes on K{Z, 3) and, therefore, on K{T}, 3) for all /: > 1 (see [4,5]). This leads to a solution of both Alexandroff Problem 0.8(a) and the Cell-like Mapping Problem 2.5 (see [27]): 5.1. DRANISHNIKOV' S THEOREM. There is an infinite-dimensional compactum X of integral cohomological dimension 3. It remained to determine if there is an infinite-dimensional compactum X of integral dimension 2. To be able to mimic Example 4.2, one needed a continuous cohomology theory /z* so that /z*(/^(Z, 2)) = 0. While the reduced complex /iT-theory with Z/p coefficients of an Eilenberg-MacLane complex K(Z, n) vanishes for « ^ 3, the same is not true for K{7J, 2). (The reader is referred to [4] and [5] for details of these K-theoretic assertions.) The specific nature of A'-theory with Z/p coefficients (p a prime) itself plays no direct role in the analysis in [27]. The essential feature is that it is a generalized cohomology theory for which K{TJ, 3) behaves as a point; i.e., the reduced A^-theory with Z/p coefficients of K{TJ, 3) is trivial. The absence of a readily available generalized cohomology theory for which ^ ( Z , 2) is known to behave as a point requires an alternate approach to that of [27] in order to produce an infinite-dimensional compact metric space having integral cohomological dimension equal to two. In addition, the generalized cohomology theory would have to have the property that, in each dimension, its value on a finite complex is a finite group. The latter is true for ^-theory with finite coefficients. If one is asked to create a generalized cohomology theory which helps to detect that a particular compactum X of integral cohomological dimension 2 is of infinite dimension, then one has to connect this cohomology with constructing maps to S^. Namely, if dimz(X) = 2, then dim(X) = oo is equivalent to dim(X) > 3, which is equivalent to the existence of a closed subset A of X and a map / : A -^ S^ which does not extend over X. Since typical generalized cohomology theories are represented by loop spaces, Dydak and Walsh [58] decided to construct a truncated cohomology theory /i* by defining h'{X) = [X, ^-'S^] for all r ^ - 2 . As shown in [110, p. 125], for any complex L and iterated loop spaces Q^L, there is a family of functors {X —> [X, ^^L]}, associating to a space X the set of homotopy classes of maps of X to Q^^L. Restricting to m ^ 2 , the functors take values in abelian groups. Unless L happens to be an infinite loop space, these functors are not part of a generalized cohomology theory. Nevertheless, some structure is present. In particular, there is a "truncated" Mayer-Vietoris sequence that can be exploited. To be able to follow the pattern of proofs of 4.2 and 5.1, one needs to verify that /?*(/<: (Z, 2)) = 0. This is done by observing that /z*(/<:(Q, n)) = 0 for all n ^ 1, which amounts to the fact that the higher homotopy groups of S^ are finite. Now, one converts the inclusion K{Z, 2) -^ K{Q, 2) to a Serre fibration and one calculates that its fiber is a K{Q/Z, 1). If /i*(^(Q/Z, 1)) = 0, then we are done. Luckily, the last equality is a consequence of the Sullivan Conjecture proved by Miller [81]. 5.2. M I L L E R ' S THEOREM. If G is a locally finite group {i.e., each finitely generated subgroup is finite) and L is a finite-dimensional CW-complex, then the space of pointed

Cohomological dimension theory

441

maps from K(G, 1) ^ L, denoted map^(K(G, 1), L), is weakly homotopy equivalent to a point (i.e., has trivial homotopy groups). Clearly, Q/Z is locally finite. Below are the results from [58] obtained by applying the above method. 5.3. THEOREM. For m^3 and an essential map S^ -^ Q^S^, there is a compact metric space X with dimz(X) < 2, and a map X ^^ S^ such that the composition X ^^ S^ ^^ Q^S^ is essential. 5.4. COROLLARY. There exist infinite-dimensional compact metric spaces having integral cohomological dimension equal to two. 5.5. COROLLARY. There is a cell-like map / : R^ ^- 7, where dim(F) = oo. In addition to Z, one can apply this technique to other Bockstein groups, and the next result involves 'mixing' the groups used to measure cohomological dimension. The examples are, in some sense, 'minimally cohomological dimensional' but still infinite dimensional. 5.6. D Y D A K - W A L S H THEOREM. There is an infinite-dimensional compactum X with dimz(X) = 2, dimQ(X) = 1, and dimz/p(X) = 1 for each prime p. In particular, d i m z ( X x X ) = 3. One can find the details of 5.1 in [27] or [28]. The details of 5.6 can be found in [58] or in [59]. 6. Extension types and extension dimension In [107] the cohomological dimension dime {X) of a compactum was defined as the smallest integer n so that K{G, n) is an absolute extensor of X. In order for this definition to make sense one needs to show that K{G, n) e AE(X) implies K(G, ^ + 1) G AE(X) for all integers n ^ 0. This implication is not a trivial one, and the corresponding result that S" e AE(X) implies S^^^ e AE(X) is much easier to prove. These results suggest that perhaps one ought to concentrate on the underlying CW complexes S^ and K(G, n) rather than on the integer n. This was done by Dranishnikov [31], who defined a relation K ^ L for CW complexes to mean that for each compact space X, the condition K e AE(X) implies L e AE(X). This relation creates an equivalence of CW complexes, and the equivalence class [K] of K is called its extension type. The extension dimension ext-dim(Z), of a compact space Z, is defined to be the minimum of extension types of K such that K G AE(X). Thus, ext-dim(X) ^ K means that K is an absolute extensor of X. In particular, ext-dim(X) ^ S" is equivalent to the covering dimension of X being at most n, and ext-dim(X) ^ K(G, n) is equivalent to the cohomological dimension of X being at most n. In this way, the extension dimension generalizes both the covering dimension and the cohomological dimension. The cornerstone of Dranishnikov's generalized dimension theory is the following.

442

/ Dydak

6.1. DRANISHNIKOV'S DUALITY THEOREM. The extension dimension of a compact space exists, and each extension type is equal to the extension dimension of some compact space. This theorem shows that there is a duaUty between CW complexes and compact spaces. There is a natural question of characterizing compact spaces of the same extension dimension, and [31] contains a partial answer to it. 6.2. THEOREM. If X and Y are finite-dimensional compact spaces, then the extension dimensions ofXxI and Y x I coincide if and only if the dimension functions ofX and Y coincide. Dydak and Dranishnikov extended the theory of extension dimension to a larger class of spaces (so-called ext-spaces) which includes all compact spaces and all separable metrizable spaces. Also, they consider the dual relation X ^ 7 for ext-spaces which means that K G AEiY) implies K e AE{X) for all CW complexes K. This relation creates an equivalence relation of ext-spaces, and the equivalence class of X is called its dimension type and denoted by [X]. Given a CW complex K one considers the family of all dimension types [X] such that K e AE(X). If this family has a maximum, then it is called the extension universe of K and is denoted by ext-univ(^). It is shown that there is a formal duality between the concept of extension dimension and the concept of extension universe. Also, there is an informal duality between results dealing with extension types and results dealing with dimension types. Some pieces of this duality are still unknown and have been posed as open problems. Extension dimension deals with general topological spaces and, in the case of finitedimensional compacta, is related to the Bockstein algebra as seen in 6.2. There is a theory dual to extension dimension which deals with CW complexes and, in the case of countable CW complexes, one has an associated algebraic object called the dual Bockstein algebra. The whole program can be carried out for a subclass of CW complexes. The main example is the class of wedges of Eilenberg-MacLane complexes. It is shown that in this case, the major results of the cohomological dimension theory correspond to the fundamental questions in the whole program. Namely, the existence of the extension dimension corresponds to the First Bockstein Theorem and the existence of the extension universe corresponds to the Dranishnikov Realization Theorem. The Second Bockstein Theorem corresponds to the existence of the Cartesian product of dimension types and its dual corresponds to the existence of the smash product of extension types. In the remainder of this section we describe the results of [38] and [48]. In the classical definition of dimension one initially defines the relation dim(X) ^ n, which is then used to define dim(X). In the same spirit, one may talk about the relation ext-dim(X) < M without discussing the existence of ext-dim(X). 6.3. DEFINITION OF ext-dim(X) ^ M. We say that the extension dimension of a topological space X is less than or equal to M (notation: ext-dim(X) ^ M), provided the property M G A £ ( X ) holds.

Cohomological dimension theory

443

6.4. EXAMPLE. If S^ is the n-dimensional sphere and X is a normal space, then ext-dim(X) < S^ means dimX < n. Traditionally, for a normal space X, one declares dim(X) to be infinite if dim(X) ^ n does not hold for any integer n. From the point of view of extension theory it is more natural to interpret dim(X) ^ oo as 5 ^ G AE(X), where S^ = U ^ o ^ ' ' - ^"^ is a contractible CW complex and one finds it natural to consider spaces X so that any contractible CW complex K is an absolute extensor of X. 6.5. DEFINITION OF A CW-SPACE. A space X (not, in general, required to be Hausdorff) is called a cw-space provided it is a /:-space and K e AE(X) for all contractible CW complexes K. Recall that X is a ^-space if A is closed in X if and only if A fl Z is closed for all compact subsets Z of Z. As we mentioned, at certain moment a natural question of the existence of the extension dimension pops up. It turns out that one can prove its existence for ext-spaces. 6.6. DEFINITION OF AN EXT-SPACE. A cw-space X is called an ext-space if for any CW

complex K eAE{X), and any subcomplex L of K containing at most w(X) cells (w{X) is the weight of X) there is a subcomplex M containing at most w(X) cells such that M G AEiX) and LcM. Recall that the weight of a topological space X is the minimal cardinal number of elements in a basis of X. 6.7. THEOREM. The class of ext-spaces contains all compact Hausdorjf spaces and all separable metrizable spaces. 6.8. NOTATION. The class of all CW complexes is denoted by CW. The class of all cw-spaces is denoted by CW-SVACSS. The class of all ext-spaces is denoted by 8XT. The class of all compact Hausdorjf spaces is denoted by COMVACT. The class of all compact metrizable spaces is denoted by COMVACT A. The class of 2i\\ finite-dimensional compact metrizable spaces is denoted by TV-COMVACTA. The class of all separable metrizable spaces is denoted by SEVAIZABCE.

6.9. DEFINITION

OF EXTENSION TYPES AND DIMENSION TYPES.

Suppose C is a class

of topological spaces and M C CW. For simplicity we ignore the possibility that either C or M may not be a set. We define a relation ^c on M as follows: ^ ^(^ L for ^ , L e A1 if and only if ^ G AE{X) implies L G AE{X) for every X eC. Then, we define an equivalence relation K ^Q L if both K ^c L ^^^ L ^c f^ hold. An equivalence class under this relation is called an extension type or, more precisely, (C, M)-extension type. K is said to be of trivial extension

444

J. Dydak

type if XxK for all X eC. The set of all extension types is denoted by ExtTypes(C, M). Clearly ^c induces a partial order on ExtTypes(C, M). If it is not ambiguous, we denote our partial order simply by ^ . Dually, we define a relation ^^V/j on C as follows: X ^j^ F for X, F € C if and only if ext-dim(y) ^ K implies ext-dim(X) ^ K for every K e M. Then, we define an equivalence relation X ^M ^ if both X ^j^Y and X ^M^ hold. An equivalence class under this relation is called a dimension type or, more precisely, (C, M)-dimension type. X is said to be of trivial dimension type '\i XxK for all ^ G A1. The set of all dimension types is denoted by DimTypes(C, M). Clearly ^^ induces a partial order on DimTypes(C, M). If it is not ambiguous, we denote our partial order simply by ^ . 6.10. EXAMPLES. Suppose S is the set of spheres of all dimensions (including infinity). (1) 5" ^ c S"^ for any class C if n ^ m < oo, (2) if X, Y are cw-spaces, then X ^s ^ if ^nd only if dim(X) ^ dim(y), (3) let C = CO MV ACT A and n > 1. Then 5" ^ c ^ ( Z , n) but S^ ^c ^ ( Z , n) does not hold. 6.11. NOTATION. For every M G A1, the corresponding class in ExtTypes(C, M) will be denoted by [M], or simply by M if the latter is not ambiguous. Dually, for every X ^C, the corresponding class in DimTypes(C, M) will be denoted by [X], or simply by X if the latter is not ambiguous. Notice that there are minimal and the maximal elements in ExtTypes(C>V-<S7^.4C^*S, CyV^. The minimal element is generated by the two-points complex D and the maximal element is generated by the one point complex P. Given a space X we consider the class {[M] € ExtTypes(C, M) \ ext-dim(X) ^ M } . If this class has a minimal element \K\, then K is one of the best approximations of X in M. from the point of view of extension theory. 6.12. DEFINITION OF EXTENSION DIMENSION AND M C CY^, C C CYs)-SVAC£S and X is a space. If

EXTENSION UNIVERSE.

Suppose

{[M] G ExtTypes(C, M) \ ext-dim(X) ^ M\ has a minimal element, then it is called an extension dimension and denoted by ext-dim(C,yW)(X). Notice that this notation agrees with the previous notation ext-dim(X) ^ M. Moreover, if X is an ext-space, then ext-dim(X) may be interpreted as ext-dim(c w-5'P^C£:5,C W) (^) • Dually, if {[X] G DimTypes(C, M) \ ext-dim(X) ^ M\

Cohomological dimension theory

445

has a maximal element, then it is called an extension universe and denoted by ext-uniV(c,>o(M). The following proposition explains our definition of the dimension type. 6.13. PROPOSITION. Let C = CW-SPACSS and let M be the set of spheres of all dimensions {including infinity). Then (1) the dimension type [X] consists of all spaces Y of the same dimension as X, (2) the extension dimension ofX is the sphere S^^^^, and (3) the extension universe of S^ is [X]for any space X of dimension n. The following result explains the duality between the concept of extension dimension and the concept of extension universe. 6.14. DUALITY THEOREM.

(1) Let X eC. Ifexi-dim(c,M)(^) exists and equals [K], then ext-uniV((3jV/^)(^) exists and equals [X]. (2) Let K G Ai. IfQXt-um\(c,M)(^) exists and equals [X], then Qxi-dim(^c,M)(^) ^^" ists and equals [K]. The following is the most general result regarding the existence of extension dimension. 6.15. THEOREM. Suppose M C CW is closed under taking arbitrary wedges and under taking subcomplexes. IfC C CW-SVACES, then for every ext-space X the class {[M] G ExtTypes(C, M) \ ext-dim(Z) ^ M] has a minimal element. The subsequent two results deal with the existence of an extension universe. 6.16. THEOREM. For each CW complex K, there is a compact space X such that [Z] = Q'Xi-\xm\i^cOMVAC'T,CW)i^)- Moreover, if K is homotopy dominated by a finite CW complex, then X is compact metric. 6.17. THEOREM. For every countable CW complex K G M., there is a separable metrizable space X such that [K] = Qxi-dim^seVAnABCS,M)(X)Besides the formal duality between extension dimension and extension universe, there is an informal duality between extension types and dimension types. In Table 1 there are some dual statements. For further information on the topics of cohomological and extension dimension, the reader should consult A. Chigogidze's article [19] in this Handbook.

J. Dydak

446 Table 1 Compact spaces C

CW complexes M

The inverse limit of at most AT-dimensional compact spaces is at most A'-dimensional, where K e Ai.

The direct limit of CW complexes which are greater than or equal to a given extension type is greater than or equal to that type.

Every at most A'-dimensional compact space can be presented as the limit space of a cr-system of at most A'-dimensional compacta. The function ext-dim is well-defined. Countability Problem: If X is metrizable, can ext-dim(X) be represented by

If ext-dim(X) ^ K, then K is the direct limit of u;(X)-complexes Ka with ext-dim(X) ^ KaThe function ext-dim is surjective.

XeANR

Metrizability Problem: If M is countable, does there exist a metrizable X with ext-dim(X) = M? 7

dim X = n

K = S"

dime X = n

K = KiG,n)

a countable complex?

6.1. Classical cohomological dimension theory reinterpreted The purpose of this section is to show how the basic results of classical cohomological dimension theory can be interpreted using the concepts of extension type, dimension type, extension dimension, and the extension universe. Typically, there are two interpretations: one in terms of compact spaces and the other in terms of CW complexes. 6.18. DEFINITION. Let £M be the class of all possible wedges of Eilenberg-MacLane spaces K(G,n), where /i G Z+ U {oo} and G is an abelian group (if w = oo, we define A'(G, ^i) to be the one-point space). The following is a restatement of the First Bockstein Theorem in terms of CW complexes. 6.19.

THEOREM. K(G,

n) is of the same extension type as

V ^(^'^) HeoiG)

in ExtTypes(COMP^CT, £M). Here is an interpretation of the First Bockstein Theorem in terms of compact spaces. 6.20. THEOREM. Suppose X,Y eC = COMVACT. Then, (1) X ^SM ^ if and only if Dx ^ Dy, X ^SM ^ if and only if Dx = Dy, and

Cohomological dimension theory

441

(2) the extension dimension Qxt-dim(^c,SM)(^) ^f^ ^•^

y KiH^dimnX). Hea

Here is an interpretation of Bockstein functions in terms of CW complexes. 6.21. THEOREM. LetC = COMVACT. Given K e SM, consider the set T of functions D such that K ~ c V//€a ^ ( ^ ' ^ ( ^ ) ) - Then (1) T^&,and (2) inf ^ belongs to T and is a Bockstein function. The next result is an interpretation of (and is equivalent to) the Dranishnikov Realization Theorem. 6.22. THEOREM. LetC^COMVACT. Then (1) any K G £M is equivalent to the unique V/Zea ^ ( ^ ' T>{H)), where D is a Bockstein function, and (2) any K G £M. has its extension universe which is a compactum.

6.2. Dual Bockstein algebra Our next goal is to define Bockstein functions of CW complexes as the dual notion to Bockstein functions of compact spaces. In view of Theorem 6.22, one can define Bockstein functions of elements of £M in a convoluted way. We plan to define these for all CW complexes in a way dictated by duality. Since Dx(H) is defined as the minimum n with ext-dim(X) < K(H, n), the following definition makes sense. 6.23. DEFINITION OF EXTENSION FUNCTION OF A CW COMPLEX. For any CW complex L its extension function E^ : a ^- Z+ U {oo} is defined as follows: E^(H) = mm{n € Z+ U {oo} | L ^c K(H, n)}, where C = J^V-COMVACTA is the class of finite-dimensional compacta. Notice that the extension function of L depends on its extension type only. Let us explain the above definition. First, let us define K{D) for any function D. 6.24. DEFINITION fined as y Hea

OF K{D).

K{H,D{H)).

If

D

is a function on Bockstein groups, then K{D) is de-

448

J. Dydak

Here is the meaning of Definition 6.23: Consider the set T of all functions D such that L ^c K(D). Notice that D = oo belongs to T and infJT e T. As in 6.21, infJF is a Bockstein function and it is clear that it is equal to E^. 6.25. PROPOSITION. Let C =

TV-COMVACTA.

(1) IfL is simply connected, then L ^c WHea ^ ( ^ ' E^(H)). (2) IfL is a countable CW complex, then E^^ = E^ -\-\. Extension functions are numerical objects which are useful in discussing the existence of extension dimension. 6.26. (1) (2) (3)

PROPOSITION. LetXeC = TV-COMVACTA and let K beaCW complex. IfXrK, thenDx^E^. If K is simply connected, then XzK if and only if Dx ^ E^. If M C CW contains all countable Eilenb erg-MacLane complexes and [K] = ext-dim((jjV/^)(X), then E^ = Dx(4) IfK^L,thenE^ ^E^, (5) IfL e CW is simply connected, then K ^ L if and only if E^ ^ E^. (6) If D is a Bockstein function and L = K(D), then E^ = D.

One may view the definition of the wedge of Bockstein functions (see Definition 1.11) as the one that assures the identity Dxyy = Dx ^ Dy- From the point of view of CW complexes, however, one would like to achieve the identity E^^^ = E^ v E^. This forces us to introduce a new wedge operation for Bockstein functions, namely the cw-wedge of Bockstein functions. In [38] one has a formal definition of the cw-wedge. Here we give a more geometric interpretation of this definition. 6.27. DEFINITION OF THE CW-WEDGE OF BOCKSTEIN FUNCTIONS. Given a set {Da}aeA of Bockstein functions, one defines its cw-wedge V^GA ^« ^^ ^^^ extension function of the wedge V^GA ^(^a)Let us explain the sense in which the cw-wedge of Bockstein functions is dual to the wedge of Bockstein functions. 6.28. PROPOSITION. Let BT be the set of all Bockstein functions and let {Da]aeA C BT. Then, (1) VaeA ^a = inf{£: eBT\E^ Da for all a € A], and (2) V«€A ^a = sup{£ eBT\E^Da

for all a e A}.

The following is the dual to 1.14(1). 6.29.

THEOREM.

IfL = V ^ i ^/ is a countable CW complex, then

E^ = \/E^^ i^\

Cohomological dimension theory

449

In [38] there is a formal definition of the smash product of Bockstein functions. Let us define it in a more geometric way. 6.30. DEFINITION OF THE SMASH PRODUCT OF BOCKSTEIN FUNCTIONS. Suppose D\ and D2 are two Bockstein functions. Define the smash product D\ A D2 SLS the extension function of K(D\) A KiDi). Kuzminov [80] found a basis (called the Kuzminov basis) of the Bockstein algebra (BT, V, x). It is a countable subset /C of the set of Bockstein functions BT such that any element D of BJ^ can be expressed by elements of /C using the operations x and v. The first proof of the Dranishnikov Realization Theorem [27] amounted to constructing a compactum XK for each K eK, so that Dxf^ = K and 6im{XK) = d\m{K). We plan to introduce a basis of the Dual Bockstein Algebra {BJ^, V^^' A). In a sense, our basis is simpler than the Kuzminov basis and is more intuitive. Elements of our basis are defined as extension functions of Eilenberg-MacLane complexes and the Kuzminov basis can be viewed as its dual. 6.31. £^'».

DEFINITION.

For each / /

G

a and each « > 0, let E^^"" = £^(^'«). Let E^ =

The following is the dual to 1.14(2). 6.32.

THEOREM. ^KAL

6.33.

IfK and L are two countable CW complexes, then

^^K

THEOREM.

^^L

{0} U {E'^jz/ea is a basis of the dual Bockstein algebra.

The following is an interpretation of the Second Bockstein Theorem: 6.34. THEOREM. Suppose C is a class of finite-dimensional compacta and M is a class of CW complexes containing all countable Eilenberg-MacLane complexes. Then (1) ifC is closed under the Cartesian product, then the operation [X] x [F] = [X x F] is well-defined in DimTypes(C, M), and (2) ifC is closed under the smash product, then the operation [X] A [7] = [X A 7] is well-defined on non-trivial dimension types in DimTypes(C, Ai). The following presents a situation in which one can compute the extension dimension. 6.35. THEOREM. Let C = J^V-COMVACTA and let M be a class of CW complexes containing all countable CW complexes. Then (1) ifX,YeC are of non-zero dimension and Z = X x Y, then ext-dim(c,>t)(Z):

\/ '-Heo

KiH,Dz(H)

450

J. Dydak

and (2) if X,Y eC are of non-zero dimension and oo > dimF = dim// Y for all H e cr, then, ext-dim(cjV4)(^ x 7) = ext-dim(c,X)(X) A Qxi-dim(^c,M)(^)Finally, we present the situation in which having the same extension dimension is equivalent to having the same cohomological dimension (the case Z = / is in 6.2). 6.36. COROLLARY. Let C = TV-COMVACTA and let Mbea class of CWcomplexes containing all countable CW complexes. Suppose Z is a continuum such that oo > dim Z = dim// Zfor all H ea. The following two properties are equivalent: (1) ext-dim(cjV/^)(X x Z) = Qxi-dim(^c,M)(^ ^ ^) ^^ ExtTypes(C, M). (2) dime X = dimo Y for all abelian groups G. An explanation of the above result is contained in [48]. 6.37. LEMMA. Let C = TV-COMVACTA and let M be be a class ofCW complexes containing all countable CW complexes. Suppose K ^c ^- Then (a) if K is simply connected, then so is L, and (b) if K is homologically n-connected (i.e., Hi(K\ Z) = Ofor i ^ n), then so is L. The above lemma implies that the following definition makes sense.

6.38. DEFINITION OF SIMPLY CONNECTED EXTENSION TYPE. We say that the extension dimension of a finite-dimensional compactum X is simply connected (respectively, homologically n-connected) provided K is simply connected (respectively, homologically nconnected) for any CW complex K which is an absolute extensor of X. Here is an explanation of Corollary 6.36 (see [48]). 6.39. THEOREM. Let C = TV-COMVACTA and let M be a class of CW complexes containing all countable CW complexes. If X and Y are two elements of C whose extension dimensions are simply connected, then ext-dim((^j\/f)(X) = ext-dim(c 7\/^)(F) in ExtTypes(C, M) if and only if Dx = Dy. 6.40. THEOREM. The extension dimension of a compactum X is homologically n-connected if and only if Dx ^ n -\-l. 6.41.

DEFINITION OF DIMENSION ALGEBRA AND EXTENSION ALGEBRA. Let C =

TV-CO MV ACT A and let M be the class of all countable CW complexes. The dimension algebra is (DimTypes(C, M), v, x) and the extension algebra is (ExtTypes(C, M), V,A).

6.42. THEOREM. There is a natural, non-injective homomorphism 0 : (DimTypes(C, M), v, x) -^ (BJ^, v, x) from the dimension algebra to the Bockstein algebra.

Cohomological dimension theory

6.43.

THEOREM.

451

There is a natural, non-injective homomorphism

/ ^^ \ ^ : (ExtTypes(C, M ) , v, A) ^ ( BT, \J, A j from the extension algebra to the dual Bockstein algebra. Proofs of Theorems 6.1 and 6.2 can be found in [31]. All of the remaining results of this section (with the exception of 6.37-6.40) can be found in [38]. Proofs of 6.37-6.40 will be pubhshed soon in [48].

7. Extension theory In 1992, during a workshop on cohomological dimension theory in Knoxville, TN, A. Dranishnikov presented certain results (see [31] or 6.1-6.2 in this paper) of generalized dimension theory which he called "extension theory". Inspired by his talk, the author thought that perhaps extension theory could be a theory encompassing both the covering and the cohomological dimension theories. Up to that point results and conjectures in cohomological dimension theory were created by formally replacing dim with dime. The author's idea was that one ought to create results/conjectures which would formally imply the corresponding results in both dimension theory and cohomological dimension theory. Let us demonstrate this on the following basic example: Since at that time the author was working on Kuzminov's question "Does dime (A \J B) ^ dime (A) + dime (5) + 1 hold?", which was formally created from the Urysohn-Menger theorem which states that dim(A U 5) ^ dim(A) + dim(5) + 1, it was natural to try to translate the inequality dim(A U 5) < dim A + dim5 + 1 into the language of absolute extensors. A simple translation is: if 5^ G AE{A) and 5^

G AE{B),

then 5'"+«+i

G AE{A

U B).

What is the connection between S"^, S"", and 5'^+«+i ? The answer is: 5'^+«+i is the join of 5^ and 5". Using this observation, the author [47] proposed the following Conjectures 7.1-7.7 (either as specific conjectures or problems which subsequently were converted into conjectures in [53]). 7.1. CONJECTURE. \i AE(AUB).

K

e AE{A) and L e AE(B) are CW complexes, then K ^ L e

7.2. CONJECTURE. If /^ * L G AE(X), then there are subsets A, B of X such that X = AUBmdK e AE(A) and L e AE(B).

452

J. Dydak

7.3. CONJECTURE. If K e AE(X) is a countable CW complex, then there is a completion IC of X with K eAE(X). 1 A. CONJECTURE. Let /^ be a countable CW complex. Any X with K eAE{X) admits compactification X with K G AE{X) if and only if K is homotopy dominated by a finite polyhedron. 7.5. CONJECTURE. IfK, L are CW complexes and/i: eAE{X), L GA£(y), then/^ A L € AE{X X y). 7.6. CONJECTURE . Suppose ^ is a countable CW complex. There is a universal separable space in the class [X\K e AE{X)]. 1.1. CONJECTURE. Suppose ^ is a CW complex. {X\K e AE(X)] has a universal compact space if and only if K is homotopy dominated by a finite polyhedron. Several of the conjectures above were subsequently answered in the positive as indicated by the following list. (a) Conjecture 7.1 by the author [47] for arbitrary metrizable spaces A, B and arbitrary CW complexes K, L. (b) Conjecture 7.2 by Dranishnikov [34] for compact spaces X and by Dranishnikov and Dydak [39] for separable metric spaces. (c) Conjecture 7.3 by Olszewski [89]. (d) Conjecture 7.5 by Dranishnikov and Dydak [39] in the case of X and Y being finitedimensional, and Y being compact. (e) Conjecture 7.6 by Olszewski [90]. Not only do extension theory results have simpler proofs than their predecessors in cohomological dimension theory, but they also explain certain phenomena in cohomological dimension theory and are sometimes more general than expected. Let us demonstrate this in the case of a result of Dranishnikov [34] result: 7.8. THEOREM. Suppose X is a compactum and K, L are countable CW complexes. If K ^ L e AEiX), then there are subsets A, B of X such that X = AU B, K e AE(A), and LeAE{B). He applied this result successfully to the Mapping Intersection Problem in codimension different from two. It turns out that Theorem 7.8 also imphes Dranishnikov's Realization Theorem. In the process of proving Theorem 7.8, Dranishnikov generalized a useful result from classical dimension theory. 7.9. ElLENBERG-BORSUK THEOREM. If X is a separable metrizable space of dimension n, then for any map f :A ^^ S^, A closed in X and k < n, there is an extension f '.U ^^ K of f over an open set U such that dim(Z — U)^n—k—\.

Cohomological dimension theory

453

Here is Dranishnikov's generalization (see [34]). 7.10. GENERALIZED EILENBERG THEOREM. Let K, L be countable CW complexes. IfX is a compactum and K ^ L is an absolute extensor ofX, then for any map f : A^> K, A closed in X, there is an extension f :U -^ K of f over an open set U such that L € AE{X - U). This result was verified for X separable and metrizable by Dranishnikov and Dydak [39]. 7.11. GENERALIZED EILENBERG-BORSUK THEOREM. Let Lbea countable CW complex. If X is a separable metrizable space and K ^ L is an absolute extensor of X for some CW complex K, then for any map f :A-^ K, A closed in X, there is an extension f'.U^^Koff over an open set U such that L € AE{X — U). [39] also contains extension theory version of the following classical result. 7.12. THEOREM. IfX is a separable metrizable space of finite dimension n, then for any k 0. For any separable metrizable space X of finite positive dimension, there is a closed subset Y ofX with dimo, (Y) = max(dimG, (X) — k, I) for i = 1,. ..,n. When translating results from extension theory to cohomological dimension theory we find that the following theorem is of fundamental importance. 7.14. THEOREM. Suppose X is a metrizable space and K is a connected CW complex. Consider the following conditions: (1) KeAE{X). (2) SP'^{K)eAE{X). (3) dim//^^(/^;Z)(X) ^ m / o r a / / m ^ O . (4) dim;,^(/^;Z)(X) ^ m for all m^ 0. Then, (1) implies (2). IfK is simply connected, then conditions (2)-(4) are equivalent. IfX is of finite dimension and K is simply connected, then conditions (l)-(4) are equivalent. Recall that SP^(K) is the infinite symmetric product of K (see [26]). For X compact, Theorem 7.14 is due to Dranishnikov [31]. Subsequently, it was generalized to metrizable spaces by Dydak [45,47]. Theorem 7.14 is reminiscent of the classical Hurewicz Theorem, and in fact part of its proof relies on the Serre version of the Hurewicz Theorem. Thus, it represents a point of overlap between extension theory and algebraic topology. As we mentioned, cell-like maps represent a point of overlap between extension theory an geometric topology. It seems natural to investigate points of overlap of extension theory and point set topology. This

454

/. Dydak

was done by the author [49] and the next part of this section is devoted to an account of [49]. One of the most beautiful results of general topology is: 7.15. TIETZE-URYSOHN THEOREM. Given a closed subset A of a normal space X and given f \A^^ [0, 1], / extends over X. The modem way of stating Tietze-Urysohn Theorem is: 7.15^

THEOREM.

[0, 1] is an absolute extensor of all normal spaces.

A big part of the Polish School of Topology was the study of absolute extensors for metrizable spaces, the so-called AR's. In general topology a dual approach was undertaken: given a space (or a class of spaces) K, characterize spaces X such that K G AE{X). In the process of proving the Tietze-Urysohn Theorem one needs: 7.16. URYSOHN LEMMA. Given a closed subset A of a normal space X, and given f:A^S^Cl,f extends over XtoF.X^ I. 7.16 means that one ought to generalize the concept of an absolute extensor to pairs of topological spaces. 7.17. DEFINITION. Let A be a subspace of X and let L be a subspace of K. Then {K, L) e AE(X, A) (respectively, (K, L) e ANE(X, A)) provided any map / : A ^ L extends to F:X -> K (respectively, extends to F : L^ ^- ^ for some neighborhood U of A in X). K e AE(X) (respectively, K e ANE{X)) means that (K, K) e AE(X, A) (respectively, (K, K) e ANE(X, A)) for all closed subsets A of X. Which definitions/theorems of topology can be expressed using the concept of absolute (neighborhood) extensor? Here is a list: 7.18. PROPOSITION.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

X is Hausdorff if and only if S^ e ANE(X, A) for all finite subsets A ofX. X is normal if and only if S^ e ANE(X). X is collectionwise normal if and only if D e ANE(X) for all discrete spaces D. (Urysohn Lemma) If S^ e ANE{X), then (/, S^) € AE{X). {Tietze-Urysohn Theorem) S^ eANE(X) if and only ifleAE(X). A is C*-embedded in X if and only if I e AE{X, A). A is C-embedded in X if and only if R e AE{X, A). A is P-embedded in X if and only if M e AE{X, A) for all complete AR's M. A is M-embedded in X if and only if M e AE{X, A) for all AR 's M. 5" eAE{X) if and only /fdimZ ^ n, K{G, n) e AEiX) if and only if dime X^n. If A ^-^ X is a shape equivalence and A is closed in metrizable X, then P € AE(X, A) for all polyhedra P.

Cohomological dimension theory

455

(13) If A is closed in the metrizable space X, then A'-^ X is a strong shape equivalence if and only if P e AE{X, A) and P e AE(X x I, A x I U X x {0,1}) for all polyhedra P. (14) (X, A) has Homotopy Extension Property with respect to Y if and only if Y e AE(X X I,AxIUX X {0}). Can one generalize well-known theorems to the setting of extension theory? The first part of this section implies that joins of complexes are vital in the understanding of classical results of dimension theory. Therefore, it is not surprising that the simplest join of spaces, the cone (which is the join of a space with the one-point space) plays a significant role in general extension theory. The following is a generalization of the Tietze-Urysohn Extension Theorem for cones (previously known if Y is discrete). 7.19. THEOREM. Suppose Y 7^ [point] is a Hausdorff space. Then, the following conditions are equivalent: (1) Cone{Y)eAE{X). (2) {Cone{Y),Y)eAE{X). (3) YeANE(X). The following is a general version of the Homotopy Extension Theorem: 7.20. THEOREM. Suppose A is a subspace ofX and Y is a topological space. Then, the following conditions are equivalent. (1) F G A £ ( X X / , X x {0}U

Ax/).

(2) (Cone(Y), Y) eAE(X x /, X x {0} U A x / ) . To an algebraic/geometric topologist, maps into simplicial complexes (or CW complexes) are of importance. Natural problems are: 7.21. PROBLEM. Characterize spaces X such that K e ANE(X) for all simplicial complexes K. 7.22. PROBLEM. Characterize spaces X such that Kcw ^ ANE(X) for all simplicial complexes K. (Here Kcw is K equipped with the weak topology.) It is well-known that each simplicial complex K has a natural partition of unity {A,i;}j^g^(0), called barycentric coordinates. It is also known that / : A -^ AT is continuous if and only if A^; o / is continuous for all v e K^^\ Obviously, in this case, {X^j o /}^^^(0) is a point-finite partition of unity on A. This can be generalized as follows. 7.23. THEOREM. A partition of unity {as}seS on X determines a map a:X ^ c\(As) C /^. A point-finite partition of unity {as}seS on X determines a map a:X -^ As C /^. Here, As is the space of all points (xs)ses ^ ^l ^^^^ ^^^^ ^^ ~ Ofor all but finitely many s e S, Xs ^ Ofor all s, and J^seS^s = 1-

456

/. Dydak

An solution to Problem 7.21 is: 7.24. THEOREM. K G ANE(X, A) for all simplicial complexes K if and only if any pointfinite partition of unity on A extends over X. An solution to Problem 7.22: 7.25. THEOREM. Suppose X is first countable. Kcw G ANE(X, A) for all simplicial complexes K if and only if any locally-finite partition of unity on A extends over X. Naturally, one may wonder about spaces X such that any partition of unity on a closed subset A of X extends over X. It turns out that these are precisely the collectionwise normal spaces. Let us demonstrate a few additional insights one gains by using partitions of unity. Suppose X is normal and has a a-locally finite basis IJ tin • One can find a partition of unity of^ on X such that Cozero(an) contains Un- Now, a = [oin/2^}n^\ is a partition of unity on X such that Cozero{oi) is a basis of X. By interpreting a as a map a\X -^ c\{As) one easily gets that Of: X -> a{X) is a homeomorphism. In particular, X is metrizable. Thus, we get a new metrization criterion as follows. 7.26. THEOREM. X is metrizable if and only if there is a partition of unity a on X such that Cozero(a) is a basis ofX. Let Y be the convex hull of ot{X) and suppose y = c\ - a{x\) -h C2 • ^(^2) belongs to the closure of ot{X) in Y. Choose s,t e S so that x\ G Cozero{ots), X2 G Cozero{oit), and Cozeroias) fl Cozero{at) = 0. Notice that y G Cozero(Xs) H Cozero(Xt) but a{X) Pi Cozero(Xs) H CozeroiXt) = 0, a contradiction. Thus, one gets: 7.27. KURATOWSKI-WOJDYSLAWSKI THEOREM. A metrizable space embeds as a closed subset of a convex subset of a Banach space. By a similar argument one gets that all points in Qf(X) are linearly independent. 7.28. A R E N S - E E L L S THEOREM. A metrizable space embeds as a closed subset of a normed linear space so that all points are linearly independent. Here is a result which unifies three well-known theorems: 7.29. THEOREM. IfX is metrizable, then any (locally-finite, point-finite) partition of unity on a closed subset ofX extends over X. The case of arbitrary partitions of unity is closest to Dugundji's Extension Theorem (see its proof in [82, p. 35]). 7.30. DUGUNDJI'S THEOREM. A convex subset of a normed vector space is an absolute extensor of all metric spaces.

Cohomological

dimension theory

457

The case of point-finite partitions of unity corresponds to the fact that simphcial complexes are ANR's (see [82, p. 304]). The case of locally-finite partitions of unity corresponds to Kodama's Theorem [70]. 7.31. KODAMA' S metric spaces.

THEOREM.

A CW complex is an absolute neighborhood extensor of all

8. Cohomological dimension and its connections to other branches of topology/mathematics In this section we review applications of cohomological dimension theory which should help the reader in assessing its place in modem topology.

8.1. Compact group actions and Hilbert-Smith Conjecture Spaces with different cohomological dimensions for different Bockstein groups appear while considering free actions of p-adic groups on n-manifolds. 8.1. DEFINITION OF THE p-ADic GROUP. Let p be a prime number. The p-adic group Ap is the inverse limit of > Z/p"+^ -> Z/p'^ -^ > Z/p, where Z/p"+* -^ Z/p" is induced by the identity Z -^ Z. The main problem regarding actions of p-adic groups is: 8.2.

CONJECTURE.

The p-adic group Ap cannot act freely on an n-manifold.

Interest in Conjecture 8.2 is motivated by Smith's [102] generalization of Hilbert's Fifth Problem [66]. Hilbert asked whether a topological group G, the underlying space |G| of which, is a (finite-dimensional) manifold, admits the structure of a Lie group, i.e., whether IGI admits the structure of a real-analytic manifold on which group multiplication and inversion are analytic functions. Hilbert's problem was solved in the positive by von Neuman in 1933 [106] for compact groups and by Montgomery and Zippin in 1952 [87] for locally compact groups. Smith's generalization, which asks whether a compact topological group acting effectively (i.e., so that each element that is not the identity moves some point) on a finite-dimensional manifold must be a (subgroup of a) Lie group, is still unanswered. It is now known as the Hilbert-Smith Conjecture. The Hilbert-Smith Conjecture is known to be equivalent to the conjecture that a compact topological group G acting effectively on a compact finite-dimensional manifold M" cannot have arbitrarily small subgroups; moreover, one may restrict attention to the case that G is totally disconnected, so it is equivalent to the proposition that if G is zero-dimensional, then it is finite (see Section 10 of [102]). Work of Newman [86] and Smith [101] eliminates arbitrarily small torsion and reduces the problem to consideration of the /?-adic group (see Section 10 of [102]). Thus, Hilbert-Smith Conjecture is equivalent to Conjecture 8.2. So far the best result regarding the Hilbert-Smith Conjecture is due to Yang [113].

458

/. Dydak

8.3. YANG THEOREM. Suppose p is a prime and the p-adic group Ap acts freely on an nmanifold M ^ Then, (1) dimz(MVAp)=n + 2, (2) dimz/p(M7Ap) = n + l, (3) dimz/qCM'^/Ap) = « if q ^ p is a prime, and (4) dimQ(M7Ap)=n. In the case of a locally compact, Hausdorff, homologically n-dimensional space X supporting an effective Ap action Yang proved that dimz(X/A^) < « + 3. Actions of this kind are called dimension-raising. Examples of interesting dimension-raising actions of A^ on a non-manifold X with dim(X/G) = dim(X) + 1 or dim(X/G) = dim(X) + 2 were constructed in [73] and [94] (see [111,112]), and an example of a free A ^-action on a 2-dimensional cell-like set was constructed by Bestvina and Edwards. Dimension-raising actions of torsion groups on compact metric spaces were recently constructed by Dranishnikov and West [44]. 8.4. THEOREM ([44]). For each integer n ^ 3 and each prime p there is an action of G = H/Si (^/P)i ^^ ^ compact, 2-dimensional metric space X such that dim(X/G) = n\ moreover, dim(X x Z) = 3. 8.5. THEOREM ([44]). For each compact metric space Y, and each prime p, Y is the orbit space of an action ofG = Y\iZ\ (^/P)/ ^^ ^ metric compactum X with dimz/p(X) = 1.

8.2. Gromov-Hausdorff metric and integral dimension Cell-like maps and spaces of finite integral cohomological dimension show up when discussing limits of manifolds in the Gromov-Hausdorff metric. 8.6. DEFINITION OF HAUSDORFF DISTANCE. Let (Z,d) be a compact metric space. Given a subset A of Z its ^-neighborhood is denoted by Ns(A). If A, B are closed subsets of Z, then the Hausdorff distance d^(A, B) from A to 5 is defined as ^ f (A, B) = inf{e > 0 | A C NsiB), B c Ns(A)}. 8.7. DEFINITION OF GROMOV-HAUSDORFF DISTANCE. If A, B are compact metric spaces, then the Gromov-Hausdorff distance d^^iA, B) from A to 5 is defined as dcHi^, B) = inf{^^(A, B) \ Z contains isometric copies of A, B}. One can consider the set CM of isometry classes of compact metric spaces with the Gromov-Hausdorff metric. It turns out that CM is a complete space (see [64] or [91] for an exposition). Within CM one can consider a subset M(n, p) consisting of all compacta of dimension at most n and whose contractibility function is p.

Cohomological dimension theory

459

8.8. DEFINITION OF A CONTRACTIBILITY FUNCTION. Suppose X is a compact metric space of dimension at most n and p : [0, R\ -^ [0, oo) is a function such that p(0) = 0, p is continuous at 0, and p(t) ^t for all r ^ i?. We say that p is a contractibility function of X provided any map a:S^ ^^ X with k ^n and diam(Qf) < s ^ R extends to ^ : 5^^^ -^ X with diam()S) < p(e). It turns out that the closure of M(n, p) is contained in the subset consisting of compacta of integral cohomological dimension at most n. 8.9.

THEOREM

([84]). IfX e c\(M(n, p)), then X is a cell-like image ofY

G M(n,

p).

There is a partial converse to the above theorem. 8.10. THEOREM ([84]). Suppose X is a cell-like image of a compact n-manifold M". There is a contractibility function p and a path a;: [0, 1] -> cl(A^(n, p)) such that p(l) = X and p{t) is homeomorphic to M" for t <\.

8.3. Cohomological dimension of groups In this section we will discuss the relation between the cohomological dimension of a CAT(O) group and the cohomological dimension of its boundary. For a more detailed discussion see [6,37], and [35]. 8.11. DEFINITION OF COHOMOLOGICAL DIMENSION OF A GROUP. The cohomological dimension cdR(r) of a group F with respect to a ring R is sup{n I / / " ( F ; M) / 0 for some RF-module M). Here H^(F; M) is defined as H^(K(F, 1); M), where Ai is the local coefficient system generated by the action of F on M. If /? = Z is the ring of integers, then cdR(F) is denoted by c d ( r ) and is called the cohomological dimension of the group F. The geometric dimension dim(r) of F is the minimal dimension of CW complexes representing K(F, 1). If F is finitely presented and of finite geometric dimension, then cd/?(F) coincides with max{n I H " ( r ; RF) ^0}= max{« | H^{X; R) / 0}, where X is the universal covering space of K(F, 1), and H^{X\ R) is the cohomology group with compact supports, which coincides with the Cech cohomology group of the one-point compactification of X. 8.12. DEFINITION OF VIRTUAL COHOMOLOGICAL DIMENSION OF A GROUP. The virtual cohomological dimension vcd/?(r) of a group F with respect to a ring R is min{cd/?(r') | F M S a subgroup of F of finite index}.

460 8.13.

/. Dydak DEFINITION OF A COXETER GROUP. {v\v^

= \ for all veV,

A Coxeter group is a group with presentation

(UM;)'^^"''^^ = 1 for all (i;, w) e l)

where V is a finite set, / is a symmetric subset of V x V, and m : / ^- Z+ is a symmetric function into the set of positive integers. (F, V) is called a Coxeter system. If m = 2, then F is called a right-angled Coxeter group. It is known that Coxeter groups have finite virtual cohomological dimension. 8.14. DEFINITION OF C A T ( O ) GEODESIC SPACE. Let (X, d) be a metric space such that for every two points x,y eX there is an isometry g : [0, d(x, y)] -> X (g is called a geodesic) so that g(0) = X and g(d{x,y)) = y. Given three points x/ G Z, / = 1, 2, 3, choose an isometry i :{x\,X2,xi,} -^ R^. If, for every y lying on a geodesic joining X2 and X3, d(x\, y) < \i(x\) — y^\, where y^ is the corresponding point on the segment joining iixj) and /(X3), then X is called a CAT(O) space. 8.15. DEFINITION OF C A T ( O ) GROUP. A group F is called a CAT(O) ^rowp if the universal cover of a ^ ( r , 1) admits a CAT(O) metric so that F acts on X via isometrics. 8.16. DEFINITION OF THE BOUNDARY OF A C A T ( O ) SPACE. Given a CAT(O) space (Z, d) and given a point x e X, one considers all isometric embeddings g: [0, 00) -^ X (called geodesic rays) so that g(0) = x. The visual sphere Sx(oo) at jc is the set of all geodesic rays emanating from x with the topology of uniform convergence on all compact sets. It turns out that the topological type of all visual spheres is the same regardless of x, and this is called the boundary of (X, d). 8.17. DEFINITION OF A BOUNDARY OF A C A T ( O ) GROUP. A boundary of a CAT(O) group F is the boundary of a CAT(O) universal cover of K(F, I). In the case of a right-angled Coxeter group F there is a specific construction of the universal cover of a K(F, I) which admits a CAT(O) metric. Thus, in this case, the boundary of that specific universal cover is called the boundary of F. Now we arrive at the connection between the cohomological dimension of groups and the cohomological dimension of compacta. 8.18. B E S T V I N A - M E S S THEOREM. Suppose F is a finitely presented CAT(O) group of finite geometric dimension, and let R be a ring with unity. Then, dimR(dF) = cdR(F) — 1. This result was used by Dranishnikov to construct interesting examples of boundaries of CAT(O) groups. 8.19. THEOREM. Let T be the family of additive groups {Q} U |Jp ^^-^^{Z/p}. For every function d'.T ^^ {2, 3} such that d(Q) = 2 and d(Z/p) = 2 for all but finitely many p, there is a Coxeter group F with vcd(F) = 3 and d{G) = vcdo (F) for all G eT.

Cohomological dimension theory

461

8.20. COROLLARY. For every prime p there is a CAT(O) group Fp whose boundary dF^ is 2-dimensional, dimz/pCSTp) = 2, dimqCSTp) = 1, anJdimz/q(9rp) = \ for every prime

Further information on CAT(O) spaces and CAT(0) groups may be found in the chapters of this book by Cannon [17] and Davis [22].

8.4. Mapping intersection problem Recently there was much research done on the so-called Mapping Intersection Problem (see Conjecture 8.22). We recommend [34] as a source for the latest results. 8.21. DEFINITION OF STABLE/UNSTABLE INTERSECTION OF MAPS. Suppose f:X -^ R^ and ^ : 7 ^- /?" are two maps of compacta into Euclidean n-space. If there exists ^ > 0 such that f'{X)f^g\Y) ^ 0 for all / ^ Z ->/?", g^ F ^ /?" satisfying \f-f\ < e, \g — g^\ < s, then / and g are said to have stable intersection. If such e does not exist, then / and g are said to have unstable intersection. 8.22. CONJECTURE. Suppose X, Y are compacta. There is a pair of maps f .X -^ R^ and g.Y ^^ R^ having stable intersection if and only if dim(X x Y)^n. 8.23. DRANISHNIKOV'S THEOREM. Suppose f-.X-^R"" and g:Y ^ R"" is a pair of maps of compacta such that dim(X) 0 there exist maps f:X^ R"", g'.Y ^ R"" satisfying \f - f'\ < e, \g - g'\ < e, and dim(/^(X) H dimg\Y))^dim(XxY)-n. Theorem 8.23 represents the latest achievement in a series of results aiming at solving Conjecture 8.22. The main ingredient distinguishing it from the preceding research is that it applied Theorem 7.8.

8.5. Rings of complex and real functions on compact metric spaces Given a compactum X one can consider algebraic properties associated with the ring C R ( X ) (respectively, Cc(X)) of all real-valued (respectively, complex-valued) continuous functions on X. It turns out that certain properties of rings Mn ( C R ( X ) ) of n X n-matrices over C R ( Z ) can be expressed in terms of the rational dimension of X. 8.24. DEFINITION OF POWER SUBSTITUTION PROPERTY. A ring R with unity has the power substitution property provided a - x -\- b - y = I implies the existence of n so that a- ln-{-b • Mn (R) contains a unit. If n is the same for all choices ofa,b, then we say that R has the n-power substitution property.

462

J. Dydak

8.25. DEFINITION OF WEAK POWER SUBSTITUTION PROPERTY. A ring R with unity has the weak power substitution property provided a • R -\-1 = R fox some ideal I of R impHes the existence of n so that a - In-\-b - Mnil) contains a unit. For general rings, the power substitution property implies the weak power substitution property. In the case of commutative rings, they are equivalent. 8.26. THEOREM ([33]). Let X be a compact metric space. The following conditions are equivalent: (1) M ^ ( C R ( X ) ) has the weak power substitution property for some n ^3. (2) dimQ(X) ^ 3. (3) Mfi ( C R ( X ) ) has the weak power substitution property for all n. S.21. THEOREM ([33]). Let X be a compact metric space. The following conditions are equivalent: (1) Mn (Cc (X)) has the weak power substitution property for some n. (2) dimQ(X) ^ 1. (3) Mfi (Cc (X)) has the weak power substitution property for all n. 8.28. THEOREM ([33]). For every n and every prime p, there is an n-dimensional compactum X so that Cc(X) has the p^~^ -power substitution property.

8.6. Finitistic spaces The classical cohomological methods in the study of group actions were applied either to compact Hausdorff spaces or paracompact spaces of finite cohomological dimension (see [9] and [13]). Swan [105] introduced the concept of finitistic spaces and obtained results generalizing classical Smith-type fixed point theorems. 8.29. DEFINITION OF FINITISTIC SPACES. A space X is finitistic if every open cover of X has an open refinement of finite order. Recall that the order of an open cover is at most n if every w + 2 distinct elements of the cover have empty intersection. Since finite open covers have finite order, compact Hausdorff spaces are finitistic. Also, paracompact spaces of finite covering dimension are finitistic. Indeed, dim(X) < n, where O^n
Cohomological dimension theory

463

a typical result on toral actions and rational coefficients involves the assumption that both the total space X and the orbit space X/ T" ( r " being the n-torus) are finitistic. In an effort to weaken those assumptions, Deo and Tripathi proved the following. 8.30. THEOREM ([25]). Suppose a compact Lie group G acts on a paracompact, finitistic space X. Then, the orbit space X/ G is finitistic. The converse of that statement was proved by Deo and Singh. 8.31. THEOREM ([23]). Suppose a compact Lie group G acts on a paracompact X. If the orbit space XjG is finitistic, then X is finitistic, too. Another branch of topology where finitistic spaces surfaced recently is the cohomological dimension theory. Namely, Rubin and Schapiro obtained the following. 8.32. THEOREM ([97]). Suppose X is a paracompact, finitistic space and G is a finitely generated, abelian group. Then (a) dimG(X) = drnxcifiX), where ^X is the Cech-Stone compactification ofX, and (b) if X is metric, separable, finitistic space, then X has a metric compactification preserving cohomological dimension. Earlier, Dranishnikov [27] found a metric, separable space X so that dimz(X) = 4 and dimz()SX) > 4, and Dydak and Walsh [56] found a metric, separable space X so that dimz(X) = 4 and dimz(A:X) > 4 for any compactification /cZ of X. In a recent paper (see [52]) Dydak, Mishra, and Shukla generalized the above mentioned results of [25,23], and [97]. This was achieved by applying the methodology of cohomological dimension theory in the form of approximate liftings (see Definition 3.2). One can speciaHze approximate liftings to inclusion of skeleta as follows. 8.33. DEFINITION OF /<:-APPROXIMATIONS. Let /<: be a metric simplicial complex. A map g'.X -^ K is a K-approximation of f :X ^ K provided for each simplex A of K and each x e X, f{x) e A implies g(x) e A. g is 2in n-dimensional (respectively, finite-dimensional) K-approximation of / if it is a A^-approximation and g(X) C K^'^^ (respectively, g(X) C K^^^ for some m). Using ^-approximations one obtains a useful characterization of finitistic spaces. 8.34. THEOREM ([52]). For a paracompact space X the following conditions are equivalent. (a) X is finitistic. (b) For any metric simplicial complex K every map f .X -> K has a finitedimensional K-approximation g. (c) For any metric simplicial complex K and every m ^ —I, every map f :X ^ K has a finite-dimensional K-approximation g so that g\f~\K^"^^) = f\f~\K^^^).

464

J. Dydak

The above characterization can be used to prove another interesting characterization of finitistic spaces. 8.35. THEOREM ([65] and [52]). A paracompact space X is finitistic if and only if there is a compact subset Z of X so that X — U is finite-dimensional for every open neighborhood UofZinX. 8.36. COROLLARY. Suppose A is finitistic and a closed subset of a paracompact space X. IfX — U isfinitisticfor every open neighborhood U of A in X, then X is finitistic. 8.37. THEOREM. Suppose X is a separable metric space and A is a finitistic subset ofX. There is a Gs-subset BofX containing A so that B is finitistic. In this section we generahze some results of classical dimension theory on maps which raise or lower covering dimension. The following two fundamental theorems can be found in [63, pp. 196-200]. 8.38. THEOREM ON DIMENSION-RAISING MAPPINGS FOR DIM. Suppose f:X^Y is a closed, surjective map of normal spaces such that there is an integer k with f~\y) containing at most k elements for all y eY. Then, dim(y) ^ dim(X) -\-k — I. 8.39. THEOREM ON DIMENSION-LOWERING MAPPINGS FOR DIM. Suppose f:X^Y is a closed, surjective map of a normal space X onto a weakly paracompact normal space Y such that there is an integer k with dim(/~*(y)) < k for all y e Y. Then, dim(X) ^ dim(F) + k. First, we generalize the well-known fact that if / : Z -> F is perfect and Y is compact, then X is compact: 8.40. f~\y)

Suppose f :X ^ Y is a closed map of paracompact spaces such that isfinitisticfor all yeY.IfYis compact, then X is finitistic.

THEOREM.

One cannot weaken the assumptions of Theorem 8.40 by assuming that Y is finitistic rather than compact. Indeed, let Q be the Hilbert cube. The projection / : Q x Z ^- Z has compact fibers, Z being discrete is finitistic, and Q x Z is not finitistic (use Theorems 2.4 or 0.6). However, we may put an additional restriction on the fibers of / as seen in the next result. 8.41. THEOREM. Suppose f :X ^^ Y is a closed map of paracompact spaces such that / ~ ' ( A ) is finite-dimensional for all finite-dimensional closed subsets A ofY. If B is a closed, finitistic subset of Y, then f~\B) is finitistic. Here is a generalization of Theorem 8.39. 8.42. COROLLARY. Suppose f : X ^ Y is a closed map ofparacompact spaces such that there is an integer k^O with dim(/~' (_y)) ^ kfor allyeY.IfY is finitistic, then so is X.

Cohomological dimension theory

465

As an application we get a generalization of a result of Deo and Singh [23] (see Theorem 0.4) regarding compact Lie group actions. The generalization applies to all compact group actions. 8.43. COROLLARY. Suppose a compact, finite-dimensional group G acts on a paracompact space X. IfX/G isfinitistic, then so is X. Notice that the converse implication (i.e., if X is finitistic, then X/G is finitistic), known to be true for Lie groups G (see [25] and Theorem 8.30), does not hold for general compact and finite-dimensional groups G. A counterexample to that implication can be easily constructed using Theorem 8.4 of Dranishnikov and West. Indeed, X = 0 ^ i X„ (the discrete sum of all Xn) is 2-dimensional and admits an action of G so that X/G is not finitistic. Let us show how to reduce the result of Deo and Tripathi (see Theorem 8.30) to the finite-dimensional case. 8.44. THEOREM. Suppose G is a compact topological group. The following conditions are equivalent: (a) For any action ofG on a finite-dimensional, paracompact space X, the orbit space X/G is finite-dimensional. (b) For any action of G on a finitistic, paracompact space X, the orbit space X/ G is finitistic. Since compact Lie groups are known to satisfy condition (a) of Theorem 8.44, Theorem 8.30 of Deo and Tripathi [25] follows. 8.45. THEOREM. Suppose f :X ^ Y is a closed map of paracompact spaces such that f(A) isfinite-dimensionalfor all finite-dimensional closed subsets A ofX. IfB is a closed, finitistic subset of X, then f{B) isfinitistic. The following is a generalization of Theorem 8.38. 8.46. COROLLARY. Suppose f'.X^^Yisa closed, surjective map of paracompact spaces such that there is an integer k with f~\y) containing at most k elements for all yeY.IfX isfinitistic, then so is Y. The following is a generalization of Theorem 8.32. 8.47. THEOREM. Suppose X is a paracompact, finitistic space and K is a metric simplicial complex which is homotopy equivalent to a CW complex L whose skeleta are finite. If K is an absolute extensor ofX, then K is an absolute extensor of the Cech-Stone compactification pX of X. If K is an absolute extensor of fiX, and is complete, then K is an absolute extensor ofX.

466

/ Dydak

9. Open problems This section is devoted to some open problems in extension theory. There is one case of the Cell-Uke Mapping Problem which remains unsolved in spite of considerable effort. 9.1.

PROBLEM.

Can cell-like maps on closed 4-manifolds raise dimension?

It is known that ANR's of finite integral cohomological dimension are finite-dimensional. Therefore, specializing Problem 0.8 to ANR's makes sense only in case of G / Z. It turns out that the case G = Q is the most important one. 9.2. PROBLEM ([27]). Is there an ANR of infinite dimension but of finite rational cohomological dimension? More generally, one may seek analogs of the Dranishnikov Realization Theorem for ANR's. 9.3. PROBLEM ([27]). Characterize dimension functions of compact (respectively, separable) ANR's. In the case of extension dimension/types the following four problems seem relevant. 9.4. PROBLEM ([38]). Characterize CW complexes whose extension universe has a compact metric representative. 9.5. PROBLEM ([38]). Characterize compacta whose extension dimension has a representative being a countable CW complex. 9.6. PROBLEM ([38]). Characterize CW complexes whose extension universe has a representative being a compact ANR. 9.7.

PROBLEM

([38]). Let M be the class of all countable CW complexes.

(1) Given X e S8VAJZABCE,

is there a minimum in

{[M] G ExiTy^Q^{S£VAnABC£, (2) Given X e COMVACTA,

M) \ ext-dim(X) < M } ?

is there a minimum in

{[M] G ExtTypes(CC>M'P^CT^, M) \ ext-dim(X) ^ M } ? Conjecture 7.7 remains unsolved. Here are the most important special cases of 1.1: 9.8. PROBLEM ([109]). Let n > 1. Is there a universal space in the class of compacta of integral cohomological dimension nl

Cohomological

dimension theory

467

9.9. PROBLEM. Let « > 0. Is there a universal space in the class of compacta of rational cohomological dimension nl 9.10. PROBLEM. IS there a universal space in the class of compacta of rational cohomological dimension 1 and covering dimension 2? It is known that, in general, one cannot compactify a metric, separable space of integral dimension 4 without increasing the integral cohomological dimension. It would be of interest to resolve the following question: 9.11. PROBLEM ([56]). Suppose Z is a metric, separable space of finite integral dimension. Is there a compactum Y containing X so that the integral dimension of Y is finite?

The next two problems deal with the cohomological dimension of Coxeter groups: 9.12. PROBLEM ([37]). Characterize dimension functions d which can be realized by Coxeter groups. Is there a Coxeter group Fn so that vcdQ(r^) = 2 and vcd(r„) = n for each larger? 9.13. PROBLEM ([37]). Suppose n^S.ls 1 and dim(9r„) = nl

there a Coxeter group Fn so that dimQ(9r^) =

References [1] RS. Alexandroff, Dimensionstheorie, ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161-238 (in German). [2] RS. Alexandroff, Einige Problemstellungen in der mengentheoretischen Topologie, Mat. Sbor. 43 (1936), 619-634 (in German). [3] RS. Alexandroff, Introduction to Homological Dimension Theory and General Combinatorial Topology, Nauka, Moscow (1975) (in Russian). [4] D.W. Anderson and L. Hodgkin, The K-theory of Eilenberg-MacLane complexes. Topology 7 (1968), 317-329. [5] V.M. Buchstaber and A.S. Mischenko, K-theory on the category of infinite cell complexes. Math. USSR Izv. 32 (2) (1968), 560-604. [6] M. Bestvina and G. Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), 469481. [7] B.F. Bockstein, Homological invariants of topological spaces I, Trudy Moskov. Mat. Obshch. 5 (1956), 3-80 (in Russian; English translation in Amer. Math. Soc. Transl. 11 (3) (1050)). [8] B. Booss and D.D. Bleecker, Topology and Analysis, Springer, Berlin (1985). [9] A. Borel et al.. Seminar on Transformation Groups, Ann. Math. Studies, Vol. 46, Princeton University Press, NJ (1960). [10] V.G. Boltyanskij, An example of two-dimensional compactum whose topological square is threedimensional, Dokl. Akad. Nauk SSSR 67 (1949), 597-599. [11] V.G. Boltyanskij, On dimensional full-valuedness of compacta, Dokl. Akad. Nauk SSSR 67 (1949), 773777. [12] K. Borsuk, Un theoreme sur la prolongements des transformations. Fund. Math. 29 (1937), 161-166. [13] G.E. Bredon, Introduction to compact transformation groups. Academic Press, New York (1972). [14] G.E. Bredon, Sheaf Theory (2nd edition). Springer, New York (1997).

468

/

Dydak

[15] G.E. Bredon, F. Raymond and R.F. Williams, p-Adic groups of transformations. Trans. Amer. Math. Soc. 99 (1961), 488-498. [16] L.E.J. Brouwer, Uberden naturlichen Dimensionsbegrijf, J. Reine Angew. Math. 142 (1913), 146-152 (in German). [17] J.W. Cannon, Geometric group theory. Handbook of Geometric Topology, R.J. Daverman and R.B. Sher, eds, Elsevier, Amsterdam (2002), 261-305. [18] A. Chigogidze, Cohomological dimension of Tychonoff spaces. Topology Appl. 79 (1997), 197-228. [19] A. Chigogidze, Infinite-dimensional topology and shape theory. Handbook of Geometric Topology, R.J. Davermann and R.B. Sher, eds, Elsevier, Amsterdam (2002), 307-371. [20] H. Cohen, A cohomological definition of dimension for locally compact Hausdorff spaces, Duke Math. J. 21 (1954), 209-224. [21] R.J. Daverman, Decompositions of Manifolds, Academic Press (1986). [22] M.W. Davis, Nonpositive curvature and reflection groups. Handbook of Geometric Topology, R.J. Daverman and R.B. Sher, eds, Elsevier, Amsterdam (2002), 373^22. [23] S. Deo and T.B. Singh, On the converse of some theorems about orbit spaces, J. London Math. Soc. 25 (1982), 162-170. [24] S. Deo, T.B. Singh and R.A. Shukla, On an extension of localization theorem and generalized Conner conjecture. Trans. Amer. Math. Soc. 269 (1982), 395-402. [25] S. Deo and H.S. Tripathi, Compact Lie group actions onfinitistic spaces. Topology 21 (1982), 393-399. [26] A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958), 239-281 (in German). [27] A.N. Dranishnikov, Homological dimension theory, Russian Math. Surveys 43 (4) (1988), 11-63. [28] A.N. Dranishnikov, On a problem of RS. Alexandroff, Math. USSR Sbor. 63 (2) (1988) AU-Aie, Mat. Sbor. 135 (4) (1988), 551-557. [29] A.N. Dranishnikov, Cohomological dimension is not preserved by Stone-Cech compactification, C. R. Bulgarian Acad. Sci. 41 (1988), 9-10 (in Russian). [30] A.N. Dranishnikov, On intersection ofcompacta in euclidean space II, Proc. Amer. Math. Soc. 113 (1991), 1149-1154. [31] A.N. Dranishnikov, Extension of maps into CW complexes. Mat. Sbom. 182 (9) (1991), 1300-1310 (Math. USSR Sbor. 74 (1993), 47-56). [32] A.N. Dranishnikov, Eilenberg-Borsuk theorem for maps into arbitrary complexes. Mat. Sbor. 185 (1994), 81-90. [33] A.N. Dranishnikov, The power substitution for rings of complex and real functions on compact metric spaces, Proc. Amer. Math. Soc. 123 (1995), 2887-2893. [34] A.N. Dranishnikov, On the mapping intersection problem. Pacific J. Math. 173 (2) (1996), 403-412. [35] A.N. Dranishnikov, On the virtual cohomological dimensions ofCoxeter groups, Proc. Amer. Math. Soc. 125(1997), 1885-1891. [36] A.N. Dranishnikov, On dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space. Trans. Amer. Math. Soc. 352 (2000), 5599-5618. [37] A.N. Dranishnikov, On boundaries of hyperbolic Coxeter groups. Topology Appl. I l l (2001), 343-353. [38] A. Dranishnikov and J. Dydak, Extension dimension and extension types, Proc. Steklov Institute of Mathematics, Vol. 212 (1996), 55-88. [39] A. Dranishnikov and J. Dydak, Extension theory of separable metrizable spaces with applications to dimension theory. Trans. Amer. Math. Soc. 353 (2001), 133-156. [40] A. Dranishnikov, J. Dydak and J. Walsh, Cohomological dimension theory with applications. Unpublished notes (1992). [41] A. Dranishnikov and D. Repovs, The Urysohn-Menger Sum Formula: An extension of the Dydak-Walsh theorem to dimension one, J. Austral. Math. Soc, Ser. A 59 (1995), 273-282. [42] A.N. Dranishnikov, D. Repovs and E.Scepin, Dimension of products with continua. Topology Proc. 18 (1993), 57-73. [43] A. Dranishnikov, D, Repovs and E. Scepin, On the failure of the Urysohn-Menger sum formula for cohomological dimension, Proc. Amer. Math. Soc. 120 (1994), 1267-1270. [44] A. Dranishnikov and J. West, Compact group action that raise the dimension to infinity. Topology Appl. 80(1997), 101-114.

Cohomological

dimension theory

469

[45] J. Dydak, Cohomological dimension and metrizable spaces. Trans. Amer. Math. Soc. 337 (1993), 219234. [46] J. Dydak, Union theorem for cohomological dimension: A simple counterexample, Proc. Amer. Math. Soc. 121 (1994), 295-297. [47] J. Dydak, Cohomological dimension and metrizable spaces, II, Trans. Amer. Math. Soc. 348 (1996), 1647-1661. [48] J. Dydak, (in preparation) (1996). [49] J. Dydak, Extension theory: The interface between set-theoretic and algebraic topology. Topology Appl. 20(1996), 1-34. [50] J. Dydak, Covariant and contravariant approaches in topology, Intemat. J. Math, and Mathematical Sci. 20 (1997), 621-626. [51] J. Dydak, Covariant and contravariant points of view in topology with applications to function spaces. Topology Appl. 94 (1999), 87-125. [52] J. Dydak, S.N. Mishra and R.A. Shukla, Onfinitistic spaces. Topology Appl. (to appear). [53] J. Dydak and J. Mogilski, Universal cell-like maps, Proc. Amer. Math. Soc. 122 (1994), 943-948. [54] E. Dyer, On dimension of products. Fund. Math. 47 (1959), 141-160. [55] J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., Vol. 688, Springer, BerUn (1978). [56] J. Dydak and J.J. Walsh, Spaces without cohomological dimension preserving compactifications, Proc. Amer. Math. Soc. 113 (1991), 1155-1162. [57] J. Dydak and J.J. Walsh, Aspects of cohomological dimension for principal ideal domains. Preprint. [58] J. Dydak and J.J. Walsh, Dimension theory and the Sullivan conjecture. Topology Hawaii, K.H. Dovermann, ed.. World Scientific, Singapore (1992), 75-89. [59] J. Dydak and J.J. Walsh, Infinite-dimensional compacta having cohomological dimension two: An application of the Sullivan Conjecture, Topology 32 (1993), 93-104. [60] J. Dydak and J.J. Walsh, Complexes that arise in cohomological dimension theory: A unified approach, J. London Math. Soc. 48 (1993), 329-347. [61] S. Eilenberg, Un theoreme de la dualite. Fund. Math. 26 (1936), 280-282. [62] R. Engelking, General Topology, Berlin (1989). [63] R. Engelking, Theory of Dimensions Finite and Infinite, Sigma Series in Pure Math., Vol. 10, Heldermann (1995). [64] K. Grove, Metric differential geometry. Differential Geometry, Proc. Nordic Summer School, Lyngby 1985, VL. Hansen, ed.. Springer Lecture Notes in Math., Vol. 1263, Springer, Berlin (1987), 171-227. [65] Y. Hattori, A note onfinitistic spaces. Questions Answers Gen. Topology 3 (1985), 47-55. [66] D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Gottingen (1900), 253-297. [67] S.T. Hu, Theory of Retracts, Wayne State University Press (1965). [68] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press (1941). [69] D.S. Kahn, An example in Cech cohomology spaces, Proc. Amer. Math. Soc. 16 (1965), 584. [70] Y. Kodama, Note on an absolute neighborhood extensor for metric spaces, J. Math. Soc. Japan 8 (1956), 206-215. [71] Y Kodama, On a problem of Alexandrojf concerning the dimension of product spaces I spaces, J. Math. Soc. Japan 10 (1958), 380-404. [72] Y Kodama, Test spaces for homological dimensions spaces, Duke Math. J. 29 (1962), 41-50. [73] A.N. Kolmogorov, Uber offene Abbildungen, Ann. of Math. 38 (1937), 488^98. [74] A. Koyama A characterization of compacta which admit acyclic UV^~ -resolutions, Tsukuba J. Math. 20 (1996), 115-121. [75] A. Koyama and R.B. Sher Approximable dimension and acyclic resolutions. Fund. Math. 152 (1997), 43-53. [76] A. Koyama and K. Yokoi, A unified approach of characterizations and resolutions for cohomological dimension modulo p, Tsukuba J. Math. 18 (1994), 247-282. [77] A. Koyama and K. Yokoi, Cohomological dimension and acyclic resolutions. Topology Appl. (to appear). [78] G. Kozlowski, Images ofANRs, Preprint (Auburn University). [79] G. Kozlowski and J.J. Walsh, Cell-like mappings on 3-manifolds, Topology 22 (1983), 147-151. [80] V.I. Kuzminov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1^5.

470

J. Dydak

[81] H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984), 39-87. [82] S. Mardesic and J. Segal, Shape Theory, North-Holland, Amsterdam (1982). [83] W.J.R. Mitchell and D. Repovs, The topology of cell-like mappings, Rediconti del Seminario della Facolta di Scienze dell'Universita di Caghari LVIII (1988), 265-300. [84] T. Engel Moore, Gromov-Hausdorjf convergence to non-manifolds, J. Geom. Anal, (to appear). [85] J. Nagata, Modern Dimension Theory, Sigma Series in Pure Math., Vol. 2, Heldermann (1983). [86] M.H.A. Newman, A theorem on periodic transformations of spaces. Quart. J. Math. 2 (1931), 1-8. [87] D. Montgomery and L. Zippin, Small subgroups of locally compact groups, Ann. of Math. 56 (1952), 213-241. [88] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Tracts in Pure and Applied Math., Vol. 1, Interscience Pubhshers, New York (1955). [89] W. Olszewski, Completion theorem for cohomological dimensions, Proc. Amer. Math. Soc. 123 (1995), 2261-2264. [90] W. Olszewski, Universal separable metrizable spaces for cohomological dimension. Topology Appl. 61 (1995), 293-299. [91] P. Petersen, Gromov-Hausdorff convergence of metric spaces. Proceedings of Symposia in Pure Math., Amer. Math. Soc. (1990). [92] L. Pontryagin, Sur une hypothese foundamentale de la dimension, C. R. Acad. Sci. 190 (1930), 1105-1107 (in French). [93] F. Raymond, Groups of transformations on manifolds, Proc. Amer. Math. Soc. 12 (1961), 1-7. [94] F. Raymond and R.F. Williams, Examples of p-adic transformation groups, Ann. of Math. 78 (1963), 92-106. [95] L.R. Rubin, Characterizing cohomological dimension: The cohomological dimension ofAUB, Topology Appl. 40 (1991), 233-263. [96] L.R. Rubin and P.J. Schapiro, Cell-like maps onto non-compact spaces offinite cohomological dimension. Topology Appl. 27 (1987), 221-244. [97] L.R. Rubin and P.J. Schapiro, Compactifications which preserve cohomological dimension, Glasnik Matematicki 28 (1993), 155-165. [98] I.A. Shvedov, On some general properties of dimension, Soviet Math. Dokl. (1968). [99] K.A. Sitnikov, Combinatorial topology of non-closed sets I, Mat. Sbor. 34 (1954), 3-54. [100] E.G. Skljarenko, On the definition of cohomology dimension spaces, Soviet Math. Dokl. 6 (1965), 478479. [101] PA. Smith, Transformations offinite period, III, Newman's Theorem, Ann. of Math. 42 (1941), 446-458. [102] PA. Smith, Periodic and nearly periodic transformations. Lectures in Topology, R. Wilder and W. Ayres, eds. University of Michigan Press, Ann Arbor, MI (1941), 159-190. [103] E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). [104] D. Sullivan, Geometric Topology, Part I: Localization, Periodicity, and Galois Symmetry, MIT Press (1970). [105] R.G. Swan, A new method infixed point theory. Comment. Math. Helv. 34 (1960), 1-16. [106] J. von Neumann, Die Einfuhrung analytischer Parameter in topologischen Gruppen, Ann. of Math. 34 (1933), 170-190. [107] J.J. Walsh, Dimension, cohomological dimension, and cell-like mappings. Lecture Notes in Math., Vol. 870 (1981), pp. 105-118. [108] T. Watanabe, A note on cohomological dimension of approximate movable spaces, Proc. Amer. Math. Soc. 123 (9) (1995), 2883-2885. [109] J. West, Open problems in infinite-dimensional topology. Open Problems in Topology, North-Holland, Amsterdam (1990). [110] G.W. Whitehead, Elements ofHomotopy Theory, Springer, Berlin (1978). [ I l l ] R.F. Williams, A useful functor and three famous examples in topology. Trans. Amer. Math. Soc. 106 (1963), 319-329. [112] R.F. Williams, The construction of certain 0-dimensional transformation groups. Trans. Amer. Math. Soc. 129(1967), 140-156. [113] C.T. Yang, p-Adic transformation groups, Mich. Math. J. 7 (1960), 201-218. [114] A.V Zarelua, Finitely-multiple mappings, Dokl. Akad. Nauk SSSR 172 (1967), 115-llS.

CHAPTER 10

Flows with Knotted Closed Orbits John Franks Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA

Michael C. Sullivan Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, USA

Contents 1. Introduction 2. Flows in boxes 3. Abstract symbolic dynamics 4. Templates 5. Knots and links 6. The Lorenz template 7. Lorenz-like templates 8. Universal templates References

473 473 480 483 485 487 491 494 496

Abstract We survey sui results concerning dynamics of flows on S^ with special attention to the relationship between dynamical invariants and invariants of geometric topology.

HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

471

This Page Intentionally Left Blank

Flows with knotted closed orbits

473

1. Introduction One of the key objects of study in the field of dynamical systems is the topological structure of the solutions of ordinary differential equations. Formally we can think of an action of the real numbers R on a manifold and one wants to study the topology of the orbits of this action. All such orbits are either injectively immersed copies of R or embedded copies of 5 ' . The latter are called periodic orbits since if we consider the flow (l)t:M ^^ M, where t eR and M is the manifold on which the flow lives then a point x e M (sometimes called an initial condition) lies on a periodic orbit if and only if 0/Q (X)=X for some /Q ^ RAn orbit of a flow is compact if and only if it is periodic. From a classical differential equations point of view (though probably not from the topologist's viewpoint) the most useful manifolds on which to investigate flows are EucUdean spaces R", or perhaps the sphere S" if it is particularly helpful to be in a compact setting. An area where this classical setting intersects a rich topological environment is the study of flows on S^ or R^ where periodic orbits can be knotted or linked. We can then ask what knot types occur as periodic orbits of a given flow (or of any flow) as well as numerous other natural questions relating the dynamics of a flow to the topology of the knots and links which occur as closed orbits of it.

2. Flows in boxes We begin by describing a very simple but remarkably rich construction (introduced by Birman and Williams [2]) to produce examples of flows with many knotted orbits. We will see that this is much more than a method of producing examples, however. Indeed, the behavior exhibited in these examples occurs quite generally in flows on S^. We want to build simple examples of flows on subsets of S^. Here is a trivial example, but one which is important as a building block. Consider the cube, or box, B = I x I x I where / = [—1, 1] which we parameterize by (x,y,s) with x,y,s e I. We consider the "flow" on B whose orbits are the line segments (x,y,s), s e I. We orient the orbits in the direction of decreasing s. And we specify a constant speed of 2 so an orbit will enter a box and remain for one unit of time before exiting. Formally 0t(x,y,s) = (x,y,s — 2t). Strictly speaking this is not a flow because the orbits will exit B after a finite amount of time (both positively and negatively). Conceptually we think of orbits entering B at the top (^ = 1), flowing downward and exiting at the bottom (5 = —1). On the sides of B the orbits lie in the boundary. This is not a very interesting example as it stands, but we can construct very interesting examples by using multiple copies of B as building blocks and attaching parts of the bottom faces of one cube to the top faces of another. It is better to refer to entering or exiting faces rather than top or bottom faces, because we will want to embed collections of attached cubes in R^ or S^ in complicated ways for which "top" and "bottom" may not be useful descriptions.

474

J. Franks and M. C. Sullivan

r(C)

c Fig. 1. The affine attaching map r.

When we attach part of the exiting face of one box to part of the entering face of another we will, of course assume that the orbit segments exiting the first at a point of attachment immediately enter the second forming a longer orbit segment crossing both boxes. We will only allow attachments of a very special form. In particular, we will make identifications using affine maps of faces which preserve the line segments in the faces parallel to the x and y axes. That is, a point (x, y, — 1) in the exiting region of box ^i may be identified with its image under an affine map, (Xx -\-aQ,X~^y -\-bo, 1), in the entering region of box B2 provided |A| > 1 and the image under this affine map of the exiting region intersects the entering region as shown in Figure 1. In particular the affine map, which we will denote r(x, j ) , must satisfy r{x,y) = (Xx + ao,X~^y -\- bo) where |A + ao| > 1, l-A + aol > 1, \X-^+bo\ < 1,

so that if C = / X /, the image under this map of C goes completely across C in the direction parallel to the x-axis, but in the direction parallel to the y-axis C goes completely across the image of C under this affine map. Parts of one exiting region may be attached to more than one entering region of one or more than one box. The regions of attachment for all entering and exiting faces must be disjoint. See Figure 2. Different attachments are permitted to use different affine maps but they must all have the form above. In particular different choices of X are permitted, but the coefficient of x (i.e., X) must always have absolute value greater than one and, as a consequence, the coefficient of y (i.e., X~^) will have absolute value less than one. In order to make interesting examples of flows on regions made up of boxes with these kinds of identifications we will need to make attachments that form a "cycle" of boxes, so it is possible for an orbit to leave a box B, pass through others, then return to B. Clearly this is necessary if we are to create a periodic orbit. It is not difficult to see that if we form a simple cycle of boxes like that in Figure 3 then there will be a single periodic orbit.

Flows with knotted closed orbits

475

Fig. 2. Attached boxes.

Fig. 3. Two boxes, one closed orbit.

It is also not difficult to see that every other orbit in the boxes depicted in Figure 3 will eventually exit the boxes in one direction or the other. The easiest way to demonstrate these facts is to consider the "return map" defined on (part of) one of the entering faces. Let C be one of these entering faces and let Do represent the subset of points in C which will return to C when the (positively oriented) orbit through

476

/. Franks and M. C. Sullivan

lili lilliillii i,..iii

1

K

II

c Fig. 4.

them is followed, and denote by RQ the points to which we return. See Figure 4. Then this return map r: DQ -^ RQ is an affine map of the type described above for attachments. Indeed it is just the composition of the affine maps for the attachments through which an orbit passes. From the form of this affine map it is easy to see that r has a unique fixed point, say (^0, yo)' Moreover, the only points (jc, y) for which r"(jc, y) is defined for all n > 0 are those with JC = JCQ while the only points for which r"(x, y) is defined for all n < 0 are those with y = yo- Clearly the fixed point of r corresponds to a periodic orbit and any other periodic orbit (if there were any) would correspond to a periodic point of the map r. We want now to investigate an analogous object but one which is more complicated than a simple cycle. Assume we have n boxes Bi,..., Bn with multiple attachments of the type described above. We will encode the box attachments in a matrix called the transition matrix A. We define it by setting Aij equal to the number of attachments of the exiting face of Bi to the entering face of Bj. The arrangement of these attachments is important, but for the moment we will ignore it. The boxes must be embedded in R^ in such a way as to realize the attachments and this requires that the embeddings be highly non-linear. In particular, it is perfectly legitimate for the exiting face of a box to be attached to the entering face of the same box. Note that the transition matrix for our simple cycle in Figure 3 is 0 1 1 0 while the one for the attached boxes shown in Figure 5 is

Given a collection of boxes attached as described above with transition matrix A we want to investigate the collection of orbits which never exit the boxes in either direction. If

Flows with knotted closed orbits

477

B,x {-1} B,x{ij

B,x {1}

B^x {-1 (a)

Fig. 5. Another two-box flow: (a) the first return map; (b) side view of boxes.

we denote this set of points by Q then it is easy to see that it is a compact set in the interior of the boxes which is invariant under the flow xj/t'-^ ^^ ^ whose orbits locally (in one box) are the arcs given by holding x and y fixed and varying t in the box co-ordinates. The orbits of this flow are parameterized so that ^^^(jc, j , ^o) = (JC, y, ^o — 0 whenever (x, y, ^o) and (jc, y, ^0 — 0 are in the same box. 2.1. The flow ft'-^ -^ ^ described above will be called the boxed flow associated to the collection of embedded boxes. DEFINITION

We will investigate this flow by considering a "return map" similar to the one described above. The name is perhaps not the best since typically points do not return under just one iteration of the map (unless the exiting face is attached to the entering face of the same box). The difference between this map and the return map discussed before is that now if there are n boxes in our construct we will consider the return map r to be defined on a subset of the union of all entering faces and having values in a different subset of the union of all entering faces. More precisely, if x is in an entering face, say of box Bt, then r{x) is defined provided the (positively oriented) orbit through x exits Bi in a region of attachment and immediately enters another box, say Bj. The value of r(x) is the point in the entering face of Bj which is the first point of this box on the positive orbit segment starting at x. The subset of an entering face on which r is defined will in general have several components each of which will be a rectangle entirely crossing that face in the direction parallel to the y-axis, like Do in Figure 4. The image of r in each entering face will in general consist

478

J. Franks and M. C. Sullivan

of several components each of which will be a rectangle entirely crossing that face in the direction parallel to the jc-axis, Hke RQ in Figure 4. Since the transition matrix A was defined by letting A/y equal the number of attachments of box Bt to Bj it is clear that Aij of the components of the domain of r in the entering face of Bt will be mapped by r to Aij of the components of the image of r in the entering face of Bj. We will denote these domain components by Dtj (k) and the range components by Rij (k), where 1 ^k ^ Aij. Note then that the restriction r : D/y (k) -> Rij (k) is an affine map which expands the horizontal or x component and contracts the vertical or y component. Let D denote the union of all the domain components, i.e., the full domain of r and let CX)

A^ f]r''iD). n=oo

Then r : A ^- A is a homeomorphism (assuming A is non-empty). Each point in A has an associated forward and backward "itinerary". This is just the sequence of rectangles Dij (k) through which the trajectory of the point travels. More formally the the forward itinerary of a point z e A is the sequence do,d\,d2, • •. where dn is the element of the collection {Dij{k)} which contains r"(z). The backward itinerary of a point z e A is defined similarly as the sequence . . . , J_2, d-\,do where dn is the element of the collection {Dij (k)} which contains r'^ (z). Combining the two provides the complete itinerary . . . , d-2, d-\,do,d\,d2,... of the point z. 2.2. Suppose z = (xo,yo,l) ^ A is in the entering face of Bi. Then the set of (x,y, 1) G Bi with the same forward itinerary as z is

LEMMA

{(xo.y,l)\ye[-hl]}, i.e., it is the interval in the entering face of Bi which passes through z and is parallel to the y-axis. Likewise the set of (x,y,l) e Bi with the same backward itinerary as z is {(jc, >^0, 1) I ^ ^ [—1, !]}• Moreover, if two points in A have the same forward (backward) itinerary then the distance between the images of these points under ^t tends to 0 as t -> oo (t -^ — oo). These line segments with the same forward (resp. backward) itinerary are called local stable manifolds, (resp. local unstable manifolds). PROOF. The affine attaching maps described above contract each line segment parallel to the J-axis by a factor of A~ ^ and expand each line segment parallel to the Jc-axis by a factor of A. It follows that if zi = (xo,y, 1) the distance between r'^ (z) andr"(zi) \sX~"\y — yo\. Hence r" (z i) is defined for all n > 0 and z and z i have the same forward itinerary. Conversely if Z2 = (^, y, 1) is in the entering face of Bi and has the same forward itinerary as z then the distance between r" (z) and r" (z2) is greater than A" |J\: — JCQI . Since this distance must be bounded independent of n we can conclude that x =xo. From the fact that llr'^(z) - r " ( z i ) | | ^ A-"|y - yol if follows that ||^(z) - ^(zi)|| tends to 0 as f ^^ 00. A similar proof shows that points with backward itinerary equal to that of z are the interval specified and that the distance between such points tends to zero under ^t as t -^ - o o . D

Flows with knotted closed orbits

479

We can also start with a possible itinerary and show that there are points which realize it. An allowable sequence ..., d-2, d-\, do, d\, d2,... of elements of the collection {D/y (k)} is one for which r(dn) intersects dn-\-{ for all n G Z. Forward and backward allowable sequences are defined similarly. Clearly a necessary condition for a sequence to be a complete itinerary is that it be an allowable sequence, since if r{dn)0 dn+\ = 0 there are no points whose trajectories enter one box through dn and the next box through dn+\. The next lemma asserts that this is also a sufficient condition. LEMMA 2.3. Suppose that . . . , J_2, d-\,do,d\,d2,. • • is an allowable sequence of elements of the collection [Dij{k)} with do C Bi. Then the set of points in the entering face of Bi with forward itinerary do,d\,d2y... is non-empty and consists of the line segment {(xQ, y, 1) I y G [—1, 1]}, for some fixed XQ. Similarly the set of points in the entering face of Bi with backward itinerary . . . , d-2, d-\,do is a line segment of the form [{x,yo,\) \ X e[—\, l]\for some yo. And the set of points in the entering face of Bi with complete itinerary . . . , d-2, d-\,do,d\,d2, •. • is the single point (xo,yo, !)• PROOF.

By induction on n it is not difficult to see that n

Wn = f] r-\di)

= {z I r\z) e di,

O^i^n}

i=0

is a rectangle in do of the form [a,b] x [—1, 1] whose width b — a is 2X~'^~K (The width of Jo is 2X~K) It then follows that the set of points in the entering face of Bi with forward itinerary do,d\,d2,... is H ^ o ^« which is a line segment of the form {(JCQ, j , 1) | y G [—1, 1]}. A similar argument shows that the set of points with backward itinerary . . . , d-2, d-\, do is a line segment of the form {(x, >^o, 1) I -^ ^ [—1, 1]} for some >'o- The set of points in the entering face of Bi with complete itinerary . . . , d-2, d-\,do,d\,d2,... must be the intersection of these two line segments, namely the point (XQ, yoA)D Clearly the itinerary sequences associated to a point in A give a great deal of information about the point. It has proven extremely valuable to abstract this concept and consider the "symbolic dynamics" associated with a transition matrix A which we describe in the next section. It is interesting that (as we shall see) a complete topological description of A and the homeomorphism r depends only on the matrix A! As a consequence the topological type of the set of orbits which remain always in the boxes depends only on this matrix. But it is important to note that while A determines the abstract topological type of this space it says very little about the embedding of this space in R^ or S^. It is really this embedding that is the subject of this article. And just as 5^ is a rather simple topological object with a very rich class of embeddings in S^, it is also the case that as abstract topological objects the flows we have described above are well understood, but their embeddings in S^ are far from understood. The flows described in this section may seem to be of a rather special nature, and to some extent that is true. Nevertheless, the exhibit very typical behavior. In [12], for example, it

480

J. Franks and M. C. Sullivan

is shown that for any smooth flow on R^ with a compact invariant set having positive topological entropy there is an invariant subset on which the flow is qualitatively equivalent to a boxed flow in a sense we define in the next section (topological equivalence). A definition of topological entropy can be found in [20]. It is a numerical invariant of the topological complexity of a flow.

3. Abstract symbolic dynamics In this section we give an abstract "symboHc" description of a class of flows and then show that any boxed flow is topologically equivalent to such a flow. Of course, we must first specify our notion of equivalence. DEFINITION 3.1. Two flows (pt'.X ^ X and V^^: F -> y are called topologically equivalent provided there is a homeomorphism H :X ^^ Y carrying orbits of 0^ onto orbits of xj/t. The flows are called topologically conjugate provided for every t and every x e X the equation H{(t)t{x)) = \l/t{H(x)) is satisfied.

Both of these are equivalence relations. From a topological point of view topological equivalence is the more appropriate notion and we will focus largely on it. Topological conjugacy is generally too rigid for our purposes. For example, the numerical value of the period of each periodic orbit is easily seen to be an invariant of topological conjugacy, from which it follows that there are uncountably many different equivalence classes. Let Abe 3.nn X n matrix of non-negative integers. We construct a finite graph F with oriented edges as follows. F has n vertices numbered 1 to ^z and F has A/y oriented edges running from vertex / to vertex j . Each edge is assumed to have unit length. We will denote these oriented edges by eij (k),l ^k ^ A/y, and denote by E the set of all edges. We want to consider the set of bi-infinite sequences of edges and give it a topology. If the finite set E is given the discrete topology and we give the product topology to oo

/ = —OO

then this space is compact, totally disconnected and perfect (assuming E has more than one element) and hence can be shown to be homeomorphic to a Cantor set. We are interested in a subset of this space of bi-infinite sequences of edges. We will denote by ZA the subset of elements {at }/ez with the property that for each / the edge at of F ends in the vertex where az+i begins. In other words at = Cpq (k) and a/+i = Cqr (k^) for some choices of /?, q, r, k and k\ Thus, subset UA consists of precisely those sequences of oriented edges which could be traced out by an infinite continuous path on the graph which always respected the orientation. The fact that £" is a set of edges of a graph is only of heuristic importance and we could equally well consider EA as bi-infinite sequences of "symbols" satisfying a finite collection of combinatorial rules about which symbols are allowed to follow which other symbols.

Flows with knotted closed orbits

481

It is a straightforward exercise to show that EA is a closed (and hence compact) subset Introducing some dynamics, we define the map

by a {{at}) = {bi} where bi = ai-\. The map a is called the subshift of finite type based on the matrix A. It is a shift because it simply shifts the bi-infinite sequences of symbols one place to the left. If any symbol were allowed to follow any other (i.e., if the matrix A is a nonzero 1 x 1 matrix and hence any symbol is allowed to follow any other symbol) then a is called a/w// shift. The "finite type" part of the name refers to the fact that there are finitely many rules (encoded in the matrix A) about which symbols may follow which other symbols. It is straightforward to see that a is continuous and we can immediately exhibit its inverse (shifting to the right), so it is a homeomorphism. There is an alternate description of UA which will be useful for us. Let F be the graph associated to A as described above with path metric assigning unit length to each edge. Let U be the space of maps from R to T which preserve arc length and take integers in R to vertices of F. We give this space the compact-open topology. It is easy to see that the map from U to UA obtained by associating to each path of E the bi-infinite sequence of edges through which it passes, is a homeomorphism. More precisely iia(t) is an element of U we define h(a(t)) to be the sequence {a/}/GZ with the property that for each / the edge of F containing a(i + 1/2) is a/. Then /z is a homeomorphism. Also it is clear that the shift map on U is given by a(a)(t) =a(t -\-1), or more precisely h is 3. topological conjugacy from a : FA ^^ ^A ^o a : U ^^ U. The most thoroughly studied subshifts of finite type are those which are irreducible. A subshift of finite type is called irreducible if the graph F described above has the property that given any two vertices on it there is an oriented path joining them. In terms of the matrix A this is equivalent to asserting that there are no / and j such that the ijih entry of A" is zero for all « > 0. A great deal is known about irreducible subshifts of finite type. See [18], for example. We mention only that any irreducible subshift of finite type has a dense orbit and has a dense set of periodic points. Associated to a subshift of finite type a -.EA^- ^A^^ can construct an abstract flow. The construction we use is called the "mapping torus" by topologists, but unfortunately is called the "suspension" by dynamicists. Let XA be the quotient space of ZA X R under the identification {cf{x),s) ^ {x,s -\-\). Equivalently, let XA be the quotient space of 17^ x [0, 1] under the identification (jc, 1) ^ ((T(X), 0). The first of these descriptions is better for describing the flow 0^ on X, because this flow is simply the quotient of the flow 0/ defined onEA ^^\)y (Pt{x^s) = {x,s -\-t). In terms of the description of the subshift of finite type as the space of maps from R to the graph F associated to the matrix A, this flow also has a nice description. Recall that the subshift map a is defined on the space U of arc-length preserving paths mapping R to F which take integers to vertices by a{a){s) =a{s -{-\). If we let A denote the space of arc-length preserving paths mapping R to F which are consistent with the orientations of the edges of F and give A the compact-open topology, then the mapping torus flow defined above is topologically conjugate to the flow on A given by 0t {GI){S) =a{s -]-t).

482

J. Franks and M. C. Sullivan

Because of the analogous properties for the subshift of finite type a, whenever the matrix A is irreducible the flow (pt on XA has a dense orbit and a dense set of periodic orbits. 3.2. A boxed flow xj/t with transition matrix A is topologically equivalent to the mapping torus flow of the subshift of finite type a : UA -^ ^A •

PROPOSITION

In fact something stronger is true. The boxed flow il/t with transition matrix A as described above and the mapping torus flow 0^ on XA constructed from the subshift of finite type a : i7^ —> EA, are topologically conjugate. Recall from Section 2 that the ij entry of the transition matrix A equals the number of components of the attaching region of the exiting face of box Bi with the entering face of box Bj. Thus if we use the matrix A to construct a graph F and the corresponding subshift of finite type a : EA^- I^A, then the vertices of F are in one-to-one correspondence with the boxes {Bi} and the edges from vertex / to vertex j are in one-to-one correspondence with the components of the attaching region of the exiting face of box Bi with the entering face of box Bj. Then if »/^^: ^ ^ ^ is the boxed flow, this gives us a map h:Q -> A which is a conjugacy from the boxed flow to the mapping torus flow for the subshift of finite type. It is defined by first considering the map j .Q ^^ F which assigns to the point (JC, y, ^) 6 Bi the point on the edge of F corresponding to the region where the orbit oi{x,y,s) exits Bi and whose distance from the beginning of that edge is (1 — s)/2. That is, its position on that edge proportional to its distance from the entering region in Bi. Then the conjugacy h is defined using the compact-open path space description of the mapping torus flow. To a point z G ^ we first consider its orbit {%{z)} in Q and then obtain a path in F. This path is arc-length preserving and hence an element of A the space of arc-length preserving infinite paths (which is given the compact-open topology). More precisely, h{z) = oi(t) where ait) is the path ji^tiz)) in F. It is immediate that PROOF.

h{iPs(z))=a(t^s)

= 0s(oc) = ^s{hiz))

so /z is a conjugacy. It is straightforward to show that h is continuous. It is only necessary to check that h is one-to-one and onto, i.e., invertible, since ^ and A are both compact. Given an element a(t) e A we wish to find h~\a). We can obtain the point Qf(0) e F, and from the edge of F we know the Bi which must contain h~\a). Also, if s is the distance of a(0) from the start of its edge, then h~\a) = (XQ, yo, \ — 2s) for some XQ and yo in the co-ordinates of the box Bi. The fact that such an XQ and yo exist and are unique can be shown as follows. The fact that the edges of G are in one-to-one correspondence with the attaching regions of the exiting and entering faces of the boxes and the fact that each such attaching region is a subset of one of the domain components {Dij (k)} means that to the path a(t) we can associate an allowable sequence . . . , d-2, d-\,do,d\,d2,..., where each di is one of the domain components {Dij(k)] and they occur in the order in which the path a(t) crosses the corresponding edges of F. According to Lemma 2.3 there is a unique point zo = (xo,yo,l) in the entering face of Bi with this itinerary. By construction h(zo)(t) is a path in G starting at the beginning of

Flows with knotted closed orbits

483

the edge containing a(0) and following the same sequence of edges as a. Then h(zo){s) = a(0) so h(zo)(t) =a(t -s). From this it is clear that h(^s(z)) = a(t) and that ^s(z) is the unique point with this property. D A corollary of this result and its proof is the following. 3.3. Orbits of a boxed flow \l/t with transition matrix A are in one-to-one correspondence with orbits of the subshift of finite type a : EA ^^ ^ A - In particular, periodic orbits of xj/t correspond to periodic orbits of cr (or, equivalently, to periodic allowable sequences of symbols associated with HA)PROPOSITION

Rather surprisingly, if we are only interested in classifying boxed flows up to topological equivalence there is a complete and easily computed answer given by the following theorem. We emphasize that this is only for classification up to topological equivalence; the analogous question for topological conjugacy is much more subtle (see [28] or [18]). 3.4 (Franks [11]). Suppose A and B are square, non-negative, integer matrices which are irreducible and ^ and

det(/ -A) = det(/ - B) and Z V ( / - A)Z" = Z ^ / ( / - 5 ) Z ^ , where n and m are the size of A and B respectively. Despite the fact that as abstract topological flows we can easily classify boxed flows up to topological equivalence, many problems remain if we try to understand how they are situated in three-dimensional space. It is the pursuit of some understanding of these embeddings that the next section considers.

4. Templates In the previous sections we described "boxed flows", a class of flows defined on subsets of R^ (but which could be easily extended to flows on all of R^). In this section we want to begin the study of how these subsets are embedded. In particular any closed orbit of a flow on R^ is a knot and any finite set of closed orbits is a link. One of our long range objectives is the study of the relationship between dynamics and the knot types of periodic orbits. It turns out that the union of the boxes in a boxed flow is not as conceptually simple for describing a flow as another construct called a "template" which we now describe.

484

J. Franks and M.C. Sullivan

DEFINITION 4.1. The template associated with a boxed flow 0s corresponding to the union of boxes X = \^ Bi is the quotient space L of X formed by collapsing each of the boxes Bi making up X to a rectangle by identifying any two points in Bi of the form (x,y\,t) with (x,y2,t). There is a well defined semi-flow (j)s induced by
Note that two points in X are identified in forming L if and only if they are in the same box Bi and they have the same forward itinerary. This is because we saw in Lemma 2.2 that the line segments we are collapsing to points are precisely the local stable manifolds consisting of points with the same forward itinerary. The semi-flow 0^ induced by
It is important to note that a boxed flow, by definition, includes an embedding of its boxes in R^ or S^. Thus if L is the template (or dual template) associated with a boxed flow then one can embed L in R^ or S^ in such a way that each point z of L lies in the interval in U Bi which is collapsed to form z- Any two embeddings of L with this property are isotopic and any of them will be called an embedding of L associated to the boxed flow. The following result is a special case of a result of Birman and WilHams [2]. It tells us that understanding the closed orbits of a boxed flow in R^ is equivalent to understanding the closed orbits of the associated embedded template semi-flow. 4.3. There is a one-to-one correspondence between the periodic orbits of a boxed flow and the periodic orbits of the induced semi-flow on the associated embedded template. Moreover, closed orbits paired by this correspondence are isotopic as embedded circles and finite sets of closed orbits matched by this correspondence are isotopic as links {embedded finite disjoint unions of circles). THEOREM

PROOF. If /?: X ^- L is the quotient map defining L then /z is a semi-conjugacy. That is, ho
Flows with knotted closed orbits

485

If L has an embedding associated with the flow, then for any point z of yo there is precisely one point Qf(z) in /z~^(z) Pi y. Sliding the point a{z) along the interval h~^{z) to z, for all points z G yo simultaneously, defines an isotopy from y to yoD

5. Knots and links A knot k is an embedding of S^ into S^ (or R^), k'.S^ -^ S^. We are only interested in smooth embeddings. A knot may be given an orientation or preferred direction. We will always use a flow to induce an orientation on our knots. It is a common abuse of notation to use the same symbol for a knot k and its image k{S^) C S^. A link of n components is an embedding of n disjoint copies of SK Two knots k\ and k2 (or two links) are equivalent if there is an isotopy of S^ that takes ^i to k2. When we talk about a knot we almost always mean its equivalence class, or knot type. Detecting knot equivalence is the primary goal of knot theory. In order to publish papers about knots, knot theorists have developed the knot diagram. This is just a projection of a knot or link into a plane such that any crossings are transverse. The crossings are then labeled as positive or negative. See Figure 6 for the convention used here. If a knot has a diagram with no crossings then it is called an unknot or less formally a trivial knot. Suppose we have a knot diagram for k and we have parameterized the planar curve. If, using polar coordinates, dO/dt > 0 for all t, then we say that the diagram represents a braiding of k or that k is in braid form. Figure 7 shows two knot diagrams for iht figure-8 knot, only one of which is braided. The reader is encouraged to demonstrate the equivalence. It is well known that any smooth knot or link is equivalent to a braid [6, Chapter 10]. In fact templates themselves can be braided, meaning that all the closed orbits are braided simultaneously [12]. If a knot k has a braid presentation in which all the crossings are positive then k is called a positive braid. Note: there exist knots which are not positive braids, but that can be presented by diagrams with only positive crossings [27]. Though a practical algorithm for detecting knot equivalence is not known, there are many useful invariants. One of special interest to us is the genus of a knot or link. Every link forms the boundary of an embedded orientable surface, called a Seifert surface [6]. Abstractly we can attach a disk along each boundary component of a Seifert surface, that is along each component of the link, producing a closed surface. The genus of the link is then defined to be the minimum genus over all such closed surfaces.

(a)

(b)

Fig. 6. (a) a positive crossing; (b) a negative crossing.

486

J. Franks and M. C. Sullivan

Fig. 7. Thefigure-8knot.

If a knot is in braid form there are formulas which allow us to compute, or at least estimate, the genus. PROPOSITION 5.1 (Bennequin [1]). Let k be a knot with genus g. Suppose k has a braid presentation on s strands with c+ positive crossings and c_ negative crossings. Then |c+ — c_ I - 5 + 1 ^ 2^ < |c+ + c_ I - 5 + 1.

A similar result holds for links. If the braid presentation of k is positive, then we have 2g = c — s -{-1 (c = c+), a fact that was proved independently in [3]. Given a two-component link /:i U /:2. the linking number of k\ with k2 is the sum of the signs of each crossing of k\ under k2 and is denoted lk(k\, k2). The linking number is a link invariant. If the components of a link / can be separated by a 2-sphere that misses the link, then we say that / is a split link. For a two-component link I = k\ Uk2, being a split link implies lk(k\, k2) = 0, though the converse is false. The last item from knot theory we review is the notion of primeness. A knot k C S^ is composite if there exists a smooth 2-sphere S^ such that S^ Hk is just two points p and q, and if y is any arc on S^ joining p to q then the knots k\ = yL}{kr\ outside of S^)

and

k2 = }/U(/:ninsideof 5^), are each nontrivial (i.e., not the unknot). We call k\ and k2 factors of k and write k = k\ #k2. We call k the connected sum of k\ and k2.lf a. nontrivial knot is not composite, then it is prime. Figure 8 gives an example. It shows how to factor the square knot into two trefoils. Trefoils are prime. It was shown by Schubert [6, Chapter 5] that any knot can be factored uniquely into primes, up to order. Note: the unknot serves as a unit.

Flows with knotted closed orbits

Left-hand Trefoil Prime

Right-hand Trefoil Prime

487

Square Knot Composite

Fig. 8. The square knot is the sum of two trefoils.

6. The Lorenz template Perhaps the simplest example of a template is the Lorenz Template. It is shown in Figure 9. It was developed by Williams as a naive model of the strange attractor apparently associated with the Lorenz equations, a 3 x 3 ODE used to study turbulent flows. See [2] and the references cited there. Although it remains unknown whether the Lorenz equations define a "Lorenz type" attractor, the existence of such attractors has been confirmed in various families of differential equations. For a brief history of this work see [20, §7.11.2] and the recent paper [8]. It is worth pointing out that boxed flows are saddle sets and not attractors. A boxed flow that would be modeled by the Lorenz template would have transition matrix

00 The reader may wish to consider how one might embed two boxes in R^ with ends attached as prescribed by this matrix in such a way that the associated template is essentially the one shown in Figure 9. (In this figure the comers on the edges of the template have been trimmed a bit for simplicity.)

Fig. 9. The Lorenz template with a trefoil orbit shown.

488

/. Franks and M. C. Sullivan

Here we shall study a little of what is known about the class of knots realized as periodic orbits of the Lorenz template's semi-flow. An important symbolic tool is the use of words to describe orbits. We label the right band x and the left band y. Then given a staring point on the branch line, we can write down a sequence of x's and y's to describe the path of the forward orbit of our point. If the orbit is periodic we shall just record the number of symbols needed to describe one circuit. Thus the orbit shown in Figure 9 is xxyxy (and turns out to be a trefoil knot). Of course, any cyclic permutation of xxyxy would give the same path. The key fact is, there is a one-to-one correspondence between the equivalence classes of finite words in two symbols under cyclic permutations and the set of periodic orbits of the Lorenz template. Similar correspondences can be set up for any template and an appropriate symbol space. This follows from the theory of subshifts of finite type and, in particular. Proposition 3.3. Knots which occur as periodic orbits in the Lorenz template are called Lorenz knots. PROPOSITION 6.1.

(1) (2) (3) (4)

All Lorenz knots are positive braids with full twists, all torus knots (defined below) are Lorenz knots, all Lorenz knots are prime, and the only split Lorenz links have either the x or y loops as a component.

We shall only prove (1), (2), and (4). Although Williams proved (3) directly in [29] it is now known that (1) implies (3); that is positive braids with a full twist are prime knots, as Williams himself had conjectured [7,26]. It is generally believed that there are examples of positive braids with full twists that are not Lorenz knots, but this has not to our knowledge been written up. The proof of Proposition 6.1 (and several others) uses what is known as the belt trick. Consider the strip with a "loop-de-loop" as show in Figure 10. As we pull the ends apart the "loop-de-loop" turns into a full twist.

Fig. 10. The belt trick.

Flows with knotted closed orbits

489

Fig. 11. Surgery on a template.

PROOF OF (1). The full twist is revealed by doing surgery on the template. We delete the two segments of flow lines that meet at the center of the branch line and start at the branch line. Then deform the resulting new template. But this new template has just the same periodic orbits as does the Lorenz template. The surgery and deformation are shown in Figure 11. The deformation uses the belt and in the last step a move we shall call the lamp shade trick. It insures that d^/dr > 0 for all orbits except the x and y orbits (which are horizontal in Figure 11. But the result holds for these two orbits trivially. Positivity is clear as only positive crossings can be realized. D

A torus knot is a knot that can be represented by a closed curve on a standardly embedded torus. A torus is standardly embedded if it is the boundary of an unknotted solid torus. All torus knots can be represented by a pair of relatively prime integers (p,q). The number p gives the number of times the knot wraps around the long way (longitudinally), while q gives the number of time the knot wraps around the torus the short way (meridionally). A torus knot can always be presented so as to have all positive or all negative crossings. We are only interested in positive torus knots. It is not hard to check that (p,q) = (q, p), but otherwise the ip,q) representation is one-to-one for positive torus knots and p,q > 0. We shall assume p 0 full twists around the torus, though the figure shows only one full twist. Then the strip splits in two with a strands staying of the side of the torus facing the reader and b strands making an extra full twist around the torus. Then the two strips come back

490

J. Franks and M. C. Sullivan

p strands

n full twists

b strands

Fig. 12. Torus knots.

p strands

b strands loop around n times

^ strands

Fig. 13. Torus knots are Lorenz.

together. One sees that q =np -{-b, and that there is a single knot as long as p and q are relatively prime. Figure 13 shows how to place the split strip onto the Lorenz template so that any torus knot can be realized as a closed orbit of the template's semi-flow. Here we have used the belt trick n -\- I times. Note that in this figure too, only the n = 1 case is shown, but the reader should be able to see how to add on extra loops of the strip about the x band of the template. D PROOF OF (4). Any two orbits whose words involve both x's and j ' s must cross, and all the crossings are positive. Thus, the linking number is not zero. D

We conclude this section by stating one of the first general results that template theory has given to the study of flows and differential equations.

Flows with knotted closed orbits

491

6.2 (Franks and Williams). A smooth flow in R^ or S^ that has a compact invariant set and positive topological entropy has infinitely many distinct knot types among its closed orbits.

THEOREM

The invariant set of a flow 0? on a manifold M is p | _ ^ ^ ^ ^ ^ / (M). We shall not define topological entropy, but only state that it a standard device for measuring "mixing". The gist of the proof is to show that any such flow must have a part of its invariant set that can be modeled by an embedded Lorenz template. Proposition 5.1 is used to show that there are closed orbits of arbitrarily high genus and thus infinity many distinct knot types can be realized as closed orbits.

7. Lorenz-like templates By Lorenz-like templates we mean one of the templates depicted in Figure 14, where there are m half twists in the x branch and n half twists in the y branch. We shall denote such a template by L(m,«). Thus L(0,0) is the Lorenz template. Notice that L{m,n) = L(n,m) via a 180° rotation. We shall at times abuse our own notation and use the symbol L(m,n) to represent the set of knots and links realizable by the closed orbits in the semi-flow on L(m, n). 7.1 (Williams). Knots in L(0, n) for n^O are positive braids with afiill twist and hence are prime. PROPOSITION

PROOF. For n = 0 we already have this result. For n^2 Figure 15 shows how to manipulate a template L(0,n) to see the full twist. This is actually a simple application of the lamp shade trick used in the proof of Proposition 6.1(1). Positivity is clear. In [17] it shown that knots on L(0,1) do indeed have a full twist presentation, but there does not appear to be a presentation of the template where all knots are simultaneously presented as positive

X-branch

Y-branch

Fig. 14. Lorenz-like templates.

492

/. Franks and M. C. Sullivan

Fig. 15. At least n half twists.

braids with full twists. Also, the result in [26] shows that having a half twist in a positive braid is enough for primeness. D PROPOSITION 7.2. There is a lower bound on the genus ofnontrivial knots in L(0, n) of g^(n-l)/2Jorn^0. PROOF. Referring again to Figure 15, suppose we have a knot with s strands. Notice that s is also just the number of y strands. If there is only one y strand, then we have an unknot. Each half twist forces s(s — l)/2 crossings. Suppose ^ > 1. Thus, since all the crossings are positive. Proposition 5.1 gives g^ (n — l)/2. D

This shows, for example, that there can be no trefoils on L(0, n) for n ^ 3, since the trefoil is a genus one knot. PROPOSITION

7.3. As sets of knots, L(0, n) cL(0,n

- 2), for all integers n.

PROOF. The proof is pictorial. Figure 16 shows how to place L(0, n) into L(0,n — 2). Again we make use of the belt trick, paying careful attention to the sign of the new full twist. D

PROPOSITION 7.4. For n <0, L(0, n) contains composite knots. PROOF. Figure 17 shows a composite knot in L(0, —1), found by Williams [29]. Its word is xxyyyyxxy and, as the reader can check, it is a connected sum of positive and negative trefoils. Figure 18 is of a composite in L(0, —2) found in [22]. The knot type is the same as before. Its word is xxxyxxxyyxyxxyyxy. These examples and Proposition 7.3 give the result. D PROPOSITION

7.5. As sets of knots L(0, ±4) is a subset o/L(0, ±1) (respectively).

Flows with knotted closed orbits

Fig. 16. L(0, n) is in L(0, n - 2).

Fig. 17. A composite knot in L(0, — 1).

Fig. 18. A composite knot in L(—2,0).

493

494

/. Franks and M. C. Sullivan

L(0,-1

i:(0,-4)

Fig. 19. 1(0, - 4 ) as a subtemplate of L(0, - 1 ) .

PROOF. Figure 19 gives the proof for the minus case. The surgery shown starts by cutting along the y orbit. This changes the invariant part of the semi-flow, but the knot type of the new y orbit is still that of the unknot. The linking number of the new y orbit with another closed orbit will be twice the linking number of the original y with the other orbit. So, the links have changed, but only a little. Any link in L(0, —4) that does not use the y orbit is isotopic to a link in L(0, — 1). The plus case is similar. The proof goes back to when one of us, as a seventh grader, was shown that when you cut a Mobius band down the middle you get a strip with four half twists in it. D

The following dichotomy has emerged. In the case n ^ 0 the templates are positive, that is all closed orbits are positive braids, while if n <0 the templates are mixed, that is they have knots which are not positive braids. In the former case all the knots are prime, while in the latter case there are composite knots. In the next section we will see that this dichotomy is even stronger.

8. Universal templates A universal template is one which contains all knots and links. It was originally conjectured in [3] that such objects could not exist. However, Christ has recently shown that they do [13]. Here we will only outline the major steps needed to show this. More recently Christ has found a template that contains all other templates as subtemplates. Step 1. Call the template shown in Figure 20 U. Define Un to be the n fold cover of U shown in Figure 21. Christ shows that for all « > 0, L^„ is a subtemplate of L^. It is worth

Flows with knotted closed orbits

495

Fig. 20. The template U.

Fig. 21. The templates Un •

noting that his method does not involve visually constructing the isotopy as above. Instead Ghrist developed a symbolic formalism for orbit-to-orbit template embeddings to show that each Un lives in U. Step 2. For every braid b there in an n such that Z? e t/«. In Figure 22 we show how a negative crossing between the second and third strand can be realized on Un. The positive crossing case is similar. By induction one can show that any braid with a single crossing is realized by a closed orbit on some Un • We can thus construct Z? on a [/^ by concatenation.

496

/. Franks and M. C. Sullivan

Fig. 22. Any crossing can be realized on Un •

In [23] it is shown that [/ is a subtemplate of L(0, —2). This, together with Propositions 7.3 and 7.5, shows that all the templates L(0, n) for « < 0 are universal. While there are examples of positive templates with composite knots, e.g., L(«, m) for n, m > 1, there always appears to be a limit on the number of prime factors. It is conjectured to be two for the example just cited. For examples (besides the Lorenz template) where such bounds have been confirmed, see [24]. CONJECTURE 8.1. IfT is a positive template, there is a number n such all periodic orbits ofT have n or fewer prime factors.

Holmes and Christ [14] have found differential equations and an open set of parameter values in which U arises. There are even electrical circuits governed by these equations. On the other hand. Holmes [16] has pointed out that positive templates arise naturally in the dynamics of forced oscillators, provided the hypothesis of hyperbolicity holds. We remark that Christ's isotopic embedding of Un in U is quite convoluted. For the figure-8 knot the number of strands in Christ's presentation runs into the millions. This is of some comfort to Sullivan and Williams, who had looked high and low for the figure8 knot in this template without success. Still, it would be interesting to find minimum representations of a given knot or link in U. Anyone care to try? The study of knotted periodic orbits and related phenomena is a growing and exciting field. The reader wanting to learn more may wish to consult the recently published book. Knots and Links in Three-Dimensional Flows [15].

References [1] D. Bennequin, Entrelacement et equations de Pfaff, Asterisque 107-108 (1983), 87-161. [2] J. Birman and R. Williams, Knotted periodic orbits in dynamical systems I: Lorenz knots. Topology 22 (1983), 47-82. [3] J. Birman and R. Williams, Knotted periodic orbits in dynamical systems II: Knot holders for fibered knots, Contemp. Math. 20 (1983), 1-59. [4] R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations 12 (1972), 173-179. [5] R. Bowen and J. Franks, Homology for zero dimensional basic sets, Ann. of Math. 106 (1977), 73-92. [6] G. Burde and H. Zieschang, Knots, Walter de Gruyter, New York (1985). [7] R Cromwell, Positive braids are visually prime, Proc. London Math. Soc. 67 (3) (1993), 384-424. [8] F. Dumortier, H. Kokubu and H. Oka, A degenerate singularity generating geometric Lorenz attractors, Ergodic Theory Dynamical Systems 15 (5) (1995), 833-856.

Flows with knotted closed orbits

497

[9] J. Franks, Homology and Dynamical Systems, Amer. Math. Soc. (1980). [10] J. Franks, Symbolic dynamics inflows on three-manifolds. Trans. Amer. Math. Soc. 279 (1) (1983), 231236. [11] J. Franks, Flow equivalence ofsubshifts of finite type, Ergodic Theory Dynamical Systems 4 (1984), 53-66. [12] J. Franks and R. WiUiams, Entropy and knots. Trans. Amer. Math. Soc. 291 (1985), 241-253. [13] R.W. Christ, Branched two-manifolds supporting all links. Topology 36 (2) (1997), 423-^38. [14] R.W. Christ and R Holmes, An ODE whose solutions contain all knots and links, to appear in Int. J. Bifurcation and Chaos (1996). [15] R.W. Christ, P. Holmes and M. Sullivan, Knots and Links in Three-Dimensional Flows, Lecture Notes in Math., Vol. 1654, Springer, Berlin (1997). [16] P. Holmes, Knot and orbit genealogies in nonlinear oscillators. New Directions in Dynamical Systems, T. Bedford and J. Swift, eds, London Math. Soc. Lecture Notes, Vol. 127, Cambridge University Press (1988), 150-191. [17] P. Holmes and R. Williams, Knotted periodic orbits in suspensions ofSmale's horseshoe: torus knots and bifurcation sequences. Arch. Rational Mech. Anal. 90 (1985), 115-194. [18] D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge Univ. Press (1995). [19] B. Parry and D. Sullivan, A topological invariant of flows on 1-dimensional spaces. Topology 14 (1975), 297-299. [20] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press Inc., Boca Raton, FL(1994). [21] S. Smale, Dijferentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 797-817. [22] M. Sullivan, Prime decomposition of knots in Lorenz-like templates, J. Knot Theory Ramifications 2 (4) (1993), 453^62. [23] M. Sullivan, Composite knots in the figure-8 knot complement can have any number of prime factors. Topology Appl. 55 (1994), 261-272. [24] M. Sullivan, The prime decomposition of knotted periodic orbits in dynamical systems, J. Knot Theory Ramifications 3 (1) (1994), 83-120. [25] M. Sullivan, A zeta-function for flows with positive templates. Topology Appl. 66 (1995), 199-213. [26] M. Sullivan, Positive braids with a half twist are prime, J. Knot Theory Ramifications 6 (3) (1997), 405-415. [27] J.M. Van Buskirk, Positive knots have positive Conway polynomials. Knot Theory and Manifolds, D. Rolfsen, ed., Lecture Notes in Math., Vol. 1144, Springer (1983). [28] R. WiUiams, Classification ofsubshifts of finite type, Ann. of Math. 98 (1973), 120-153; Errata 99 (1974), 380-381. [29] R. WilUams, Lorenz knots are prime, Ergodic Theory Dynamical Systems 4 (1983), 147-163.

This Page Intentionally Left Blank

CHAPTER 11

Nielsen Fixed Point Theory"* Ross Geoghegan State University of New York, Binghamton, NY 13902-6000, USA

Contents Introduction Part I. The basic theory 1. Geometric fixed point theory 2. Fixed point theory in algebraic topology 3. Equivalence of the two theories onENR's 4. Reducing the number of fixed points Part II. Some connections with other areas 5. The Euler characteristic and a problem about traces of idempotents 6. Gottlieb's theorem on the center of a group 7. Periodic points, zeta functions and torsion A. Periodic point theory B. Dynamics 8. Growth rate of Nielsen numbers References

* Supported in part by NSF Grant. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

499

501 502 502 504 508 509 511 511 512 514 514 516 518 520

This Page Intentionally Left Blank

Nielsen fixed point theory

501

Introduction Nielsen Fixed Point Theory combines the ideas of the Lefschetz Fixed Point Theorem with the fundamental group to produce a richer version of the Lefschetz Theory. Just as the Lefschetz theorem concerns the "Lefschetz number", L{f) G Z , associated with a suitable map f \X ^> X, Nielsen Theory produces an invariant called the "Reidemeister trace", /?(/), not a number but an element of a certain free Abelian group. In fact, R{f) depends on a base point v and a base path r from f to / ( f ) , so I will need notation later which acknowledges this; but not in this Introduction. The theory was developed during the 1930's mainly by Reidemeister [43] and his student Wecken [50]; however, some of the main ideas appeared a little earlier in the work of Nielsen [38] and that is why the theory is named in his honor. The fixed point set, Fix(/), of a map f :X ^^ X is {x e X \ f(x)=x}.lts members are iht fixed points of / . It is well-known that when X is a finite complex, L(f) is defined, and if Fix(/) = 0 then L(f) = 0; indeed, this is the contrapositive of the Lefschetz theorem. Since L ( / ) is a homotopy invariant, a better statement is: if f is homotopic to a map g such that Fix(g) = 0 then L(f) = 0. In this form, it is reasonable to ask if the converse is true: is the vanishing of L(f) sufficient to guarantee such a gl The answer is no, and analysis of what goes wrong in the "proof" leads straight to R{f) and to the theorem that, under certain hypotheses, the vanishing of Rif) is sufficient to give such a g. I prove this for a reasonably general case in Section 4. Just as with L ( / ) , there are two definitions of R(f) when Z is a compact manifold, one geometric and the other algebraic. And just as the classical Lefschetz-Hopf theorem says that the geometric definition of L ( / ) (in terms of fixed point indices) equals the algebraic definition (an alternating sum of traces), so a central theorem of Nielsen theory says that under suitable hypotheses the geometric definition of R(f), given here in Section 1, equals the algebraic definition given in Section 2; this is proved in Section 3. In summary, my aim in Part I (Sections 1-4) is to give a more-or-less complete account of what I have just outlined, the qualification "more-or-less" referring mainly to two intuitively reasonable Propositions 1.1 and 1.2 whose detailed proofs are too long for this article and are well written up in the textbooks [2] and [33]. This is the basic theory, and every topologist should know it. In Part II (Sections 5-8) I give short accounts of some connections between Nielsen Fixed Point Theory and other parts of geometric topology. Such a selection is always subjective and perhaps controversial (for what is omitted), but I think the four topics covered suggest some reasons why geometric topologists should pay attention to Nielsen theory. Not that Part II is complete! I do not do justice to the interesting new connections between Nielsen theory and low-dimensional topology. For example, I have said that the vanishing of R(f) is usually sufficient for the existence of g homotopic to / with Fix(^) = 0, but the notable exception among compact w-manifolds is the case n =2. The discovery of a counter example for maps on surfaces by Jiang [29] was one of the important events in the recent resurgence of this subject. Jiang used a representation in a braid group, but more robust methods involving surface topology were subsequently developed [32,52]. Another recent development barely touched on here is parametrized Nielsen Fixed Point Theory, especially the 1-parameter version involving an interval of maps or a circle of

502

R. Geoghegan

maps; see [5,6,14-18,26]. One aspect of this theory sees the n-parameter analog of Rif) as lying in the nth Hochschild homology group HHni^G) (where G = 7t\ (X)) and connects the subject with geometric problems obstructed by elements in the higher ^-theory of ZG. A hint of this for the "classical" case n = 0 can be found in Section 5. Nielsen theory has applications to non-linear analysis but that is outside the scope of this article; see, for example [11]. The reader interested in knowing about recent research in Nielsen Fixed Point Theory should look at the conference proceedings volumes [3,10,30,47] and, especially recommended, [35] which is the most ambitious in scope. For a different view of Fixed Point Theory circa 1980 (not specifically Nielsen theory) see the survey article [7].

Part I. The basic theory 1. Geometric fixed point theory The point 0 is a fixed point of every linear map A: R^ -^ R"; it is an isolated fixed point if and only if 1 is not an eigenvalue of A, if and only if / — A is non-singular; in which case the sign (= ±1) of the determinant of / — A is called t h e ^ index of A at 0, denoted t(A, 0). The Jordan canonical form shows that ^A, 0) is the parity of the number of real eigenvalues > 1, counted with multiplicity, so it is a crude measure of A's "expansiveness". But to generalize L{A,0) to the non-linear case, one should note instead its topological meaning: t(A, 0) is the topological degree of the map of pairs (I — A): (R'^, R" — {0}) -> (R",R"-{0}). If M is an n-manifold and jc G M is an isolated fixed point of the map / : M ^- M, the Jp index of / at jc is the degree, t(/, jc), of the map of pairs (id - hfh-^):

{h(V), h{V) - {h(x)}) -^ (R^ R" - {0}),

where U is open in M, h:U ^^W is an open embedding (a "chart"), and V is an open n-ball neighborhood of X in U such that f(V)cU.^ When M is compact and Fix(/) = {x\,...,Xr} C M is finite, the integer L(f) = Yl)=\ ^if^^j) is the Lefschetz number of / . It is not obvious from this definition that L ( / ) is a homotopy invariant, nor is it clear how to define L ( / ) when Fix(/) is infinite, or M is replaced by a non-manifold. I will address these matters, but for geometric motivation I had best discuss this nicest case first: M a compact manifold, and Fix(/) a finite subset of its interior. By grouping together the fixed points into "fp classes", and summing the fp indices of each class separately, one gets an important refinement of the integer L ( / ) as I now explain. ' In algebraic topology, a canonical isomorphism o: Hn (R", M^^ — {0}) -^ Z» called the standard orientation on E'^, is chosen once and for all. The map (translation oih{x) to O)oinclusion provides a canonical isomorphism Hn{h{V),h{V) - {h{x)]) -^ //,i(E'MR" - {0}) -^ Z. Thus both homology groups have canonical generators mapped to 1 G Z. Interpreting (id — hfh~^)^ as an endomorphism of Z, the degree of id — hfli~^ is the image of 1. It is independent of U,h, V and any particular author's choice of standard orientation.

Nielsen fixed point theory

503

In preparation, I give some definitions which make sense for any self-map f \X ^^ X of any path connected, locally connected, semilocally simply connected space. Say that x,y e Fix(/) a r e ^ equivalent if there is a path y in X from x io y such that the loop ^ ( / o ^)~^ in X »s homotopically trivial. The equivalence classes are called^ classes: they are both closed and open subsets of Fix(/) (which, in turn, is closed in X). Thus when X is compact there are only finitely many fp classes. Pick a base point v for X and let G = 7T\ (X, v). Pick a base path r in X from v to f(v). Write 0 : G ^- G for the homomorphism induced by / using v and r; i.e., if a; is a loop at V, and [co] denotes the corresponding element of G, then 0([a>]) = [ r ] [ / o <w][r]~^ The homomorphism (piG ^^ G defines an equivalence relation on G: say g\, g2eG art semiconjugate (or (t)-conjugate) if there exists g eG such that g2= gg\(t>{g)~^ - Write G^ for the set of semiconjugacy classes; in particular, when 0 = 1, the identity homomorphism, G1 is the familiar set of conjugacy classes. There is an injective function (P from the set of fp classes of / : X ^- X to the set G^; 0 maps the fp class containing x to the semiconjugacy class containing [/x(/ o /x)~^ T~' ], where /x is any path from i; to x. One checks that

R{f, V, [r]) ^ ^

K/, F^)0(F^) G ZG0.

k=\

This is a formal integer sum of semiconjugacy classes. The number of non-zero terms in this sum is called the Nielsen number of / , denoted by N(f); in other words, N(f) is the number of fixed point classes of / having non-zero fp index. I now generalize all this to maps f :Z -^ Z where Z is a path connected compact Euclidean neighborhood retract (ENR) (i.e., a retract of an open subset of a Euclidean space) and Fix(/) is arbitrary. Since ENR's are locally simply connected, the fp classes of / have already been defined. I must define the fp index. Let Z C M^ and let r : W ^- Z be a retraction where W is open in R^. Let F be an fp class of the map / : Z ^- Z, let f/ be an open set in Z containing F such that clU HFix(/) = F, and let V = r~\U). Define d:(V,V -F)-^ (R^, R^ - 0) by d(x) =x- fr(x). There is ^ > 0 such that Z lies in Bt(0), the closed ball about 0 in R^ of radius t. Then //yv(R^, R^ - Bt(0)) = Z has a chosen generator 1, as does / / A ^ ( R ^ , R ^ - {0}) = Z. T h e ^ index, L{f, F) of / at F, is the image of the generator 1 under the following composition, considered as an endomorphism ofZ: //yv(M^,R^ - 5.(0)) "^^""^"*> //yv(R^,R^ - F ) ^^""""*"> HM{V. V - F ) J*

///v(M^,M^-{0}).

504

R. Geoghegan

As in the special case above, the Reidemeister trace is

/?(/, u, [r]) ^

Y.

^(/' ^)^(^) ^ ^^^'

fp classes F

the Lefschetz number is L ( / ) = 5Zfp classes F^(/'^) ^ ^ ' ^^^ ^^ Nielsen number is A^(/) = the number of fp classes F for which i{f, F) / 0. Here are two important technical properties of the Reidemeister trace. PROPOSITION 1.1 (Homotopy Invariance). Let F \Z ^ I ^^ Z be a homotopy, let x be a path from v to FQ(V) and let co be the path F(v, •)• Then RiFo, D, [r]) = RiF\, i;, [rco]); in particular N(Fo) = N{F\). r

Let Z ^ Z' be maps. Then r restricts to a homeomorphism Fix(5 o r) -> Fix(r o s) and s

induces a bijection r# from the fp classes of ^ o r to the fp classes of r o 5. PROPOSITION 1.2 (Commutativity). Let v be a path in Z' from v' to r{v) and let jibe a path in Z from v to s(v^). Then r#(R(s o r, v, [/x(^ o v)])) = R(r o s, v\ [v(r o /x)]).

The proofs of these two propositions are complicated though elementary. For a complete exposition see [2, Chapters 4 and 6]. Of course, in the simple case of manifolds when Fix(5 o r) is a finite subset of the interior, Proposition 1.2 is easily proved. And in case Z is a manifold and F is transverse to projection Z x I -> Z, Proposition 1.1 follows from the fact that the "fixed point set of F " is a 1-manifold whose boundary is Fix(Fo) U Fix(Fi) C Z X {0,1}; see [36] or [20, p. 123] for this method. Finally, I should add that this definition of Rif,v, [r]) can be extended to all compact ANR's - see [2, Chapter 5]. NOTE. The definition of index for arbitrary (not necessarily discrete) fixed point sets in ENR's is not found in the original German papers [43] and [50], but came later, in the work of B. O'Neill, F. Browder, J. Leray and A. Dold. See the historical notes in [2].

2. Fixed point theory in algebraic topology Starting again, I will redefine R(f, v, [r]) and L ( / ) as traces in the sense of linear algebra, first on finite complexes and then on finitely dominated spaces. Let ^ be a finite connected CW complex and lei f: K ^^ K bta. cellular map. Choose an orientation for each cell of K, thereby specifying a basis for the cellular /c-chains Ck (K) with Z-coefficients. The Lefschetz number of / is L ( / ) ^ ^ ( - l ) ^ t r a c e ( A :CK/^) ^ k>0

CkiK)).

Nielsen fixed point theory

505

By a well-known trick [48, 4.7.6] this is equal to ^ ( - l ) ^ t r a c e ( / * : / f ; t ( / r ) ^ /f^(^)).2

It follows that L{f) is a homotopy invariant; in particular it does not depend on the choice of orientations for the cells of K. Choose a base vertex v for K and write G = 7TI(K, V). Let p:(K,v) -^ {K, v) denote the universal cover. For each (oriented) cell a of ^ choose a cell a of A^ so that p(a) =a. Orient a compatibly with a. Choose a cellular base path r in K from v to f(v). Let f:K-^K denote the lift of / such that f(v) = f (1) where f is the lift of r with f(0) = C; / is a cellular map. As in Section 1, write 0 : G ^- G for the homomorphism induced by / using the base path T . The usual left action of G on ^ by covering transformations makes Ck(K) into a finitely generated free left ZG-module which is turned into a free right ZGmodule in the usual way, namely if c G Q ( ^ ) and g G G, eg is the chain (g~^)#(c). The oriented /:-cells a^ (one for each /:-cell cr^ of K) form a ZG basis. The chain map fk : Ck{K) -^ Ck(K) is not always a ZG-module homomorphism, as it satisfies fk(o^g) = fk(o^)(p(g)- Nevertheless one can usefully represent fk by a square ZG matrix whose (7, /) entry is the ZG-coefficient of a^ in the chain fki^f). Write R'{f. V. [T]) = ^ ( - 1 ) ^ trace(A : Q ( f ) ^

Ck(K)).

This element of ZG depends on the choice of lifts a^ and on the choice of orientations. That is not the case with its image R(f, v, [r]) = q(R'(f, v, [r])); here q:ZG^ ZG^ is induced by the function G ^^ G^p which sends each g to its semiconjugacy class. Explicitly:^ R{f, V, [r]) ^ ^ ( - l ) ^ ^ ( t r a c e ( / ^ : Ck(K) -^ Ck(K))) e ZG^;

it is called the Reidemeister trace. The number of non-zero coefficients in this element of ZG^ is the Nielsen number of / , denoted N(f). Of course, I have already given these names to apparently different quantities in Section 1. This double use of names will be cleared up in Section 3. My immediate task is to show that R(f, v, [r]) and A^(/), as defined in this section, are homotopy invariants. Let / = [0, 1] with the usual CW complex structure: give its vertices the orientation +1 and give its 1-cell the usual orientation. Give A' x / the product CW complex structure and give each of its cells the product orientation. Let F : ^ x / -^ ^ be Often, writers use rational coefficients here because H^iK) may have torsion, but what we have written has a recognized (and obvious) meaning. ^ One should not expect an equivalent formula to be derived from /*://*(A") -> H^{K) since to imitate the method used for /* : H^{K) -> H^{K) one would need all submodules of a free ZG-module to be free, and this is rarely true.

506

R. Geoghegan

a cellular homotopy. Take (f, 0) as base point in ^ x / and pick a base path r in ^ from v to F{v, 0). Identify 7i\ (K x /, (D, 0)) with G = n\ (K, v) via the isomorphism induced by the projection p:K x I ^^ K. Write 0 : G -^ G for the homomorphism TTi {K X / , (v, 0 ) ) - ^ TTi (ii:, F ( i ; , 0 ) ) -^^^^^ TTI (/^, i;).

Let f be the lift of r to ^ which starts at i; G^^. L e t ^ : ^ x / -^^ be the Hft of F which maps (v, 0) to f(l). The chain homotopy Dk : Ck(K) -^ Q + i (A') defined on generators by Dk(ai= (-l)^^^Fk(a X /) satisfies^ (Dk^ih + a)t+iD^)(^) = Fok(^) - F u ( a ) . Clearly, FoCi)) = f(l) and F\(v) = (fa>)(l), where a;: / ^- ^ is the path F{v, •) in K. Since Dk(cfg) = Dk{o)(t){g), this leads to the matrix equation:^ D^_i0(6^) + dk+\Dk = Fok — F\k. Applying 5Il^>o(~l)^^(trace()) to the left-hand side of this equation gives zero. Thus I have proved the first part of the following proposition; I leave the parts about the Nielsen number to the reader: 2.1 (Homotopy invariance). not depend on v or [r]; N(Fo) = N(F\). PROPOSITION

R(FQ, V, [T])

= R(F\,v, [rco]); N(f)

does

For ^ G G let c^ : G -^ G denote the conjugation Cg(g) = g~^gg- Right multiplication by g induces a bijection G^ -^ Gcgocf) and hence an isomorphism pg :ZG0 -^ ZGcgo
Now suppose the map f: K ^^ K is not cellular. Again let t; be a base vertex and r a base path for / . The Cellular Approximation Theorem gives a homotopy F . K x I -^ K with FQ = f and F\ cellular. Define R{f, v, [r]) = R{F\, v, [rw]), where co = F{v, •)• By Proposition 2.1 and the fact that a homotopy between cellular maps is homotopic, rel the ends, to a cellular homotopy, this definition of R(f,v, [r]) is independent of F. Moreover, if H:f 2:1 f is di homotopy, and X = H{v, •) then /?(/, v, [r]) = /?(/, v, [rA]). In this sense, the Reidemeister trace depends only on the homotopy class of / . "* Let £/ e be the face of the cube /^ = [— 1,1]^ obtained by holding the /th coordinate fixed at f = ± 1 . Define the corresponding incidence numbers [/^ : Ei _\] = (—1)' = —[/^ : £/,i]. At the level of cellular ^-chains d^I^ = ^i^^U^ '• Ei^]Ei^. This convention leads to the (—1)^+' in the definition of Di^{a). ^ The hnear algebra is as follows. Let /? be a ring, and (p-.R^-Ran endomorphism. If C is an /?-matrix, write (piC) for the matrix [
Nielsen fixed point theory

507

The path connected space X is finitely dominated if there exist a path connected fid

nite CW complex K and maps K^X

such that J o w is homotopic to idx• If K and X

u

have base points v and x there is no loss of generality in assuming (as I always will) that d(v) = X, u{x) = V and d ou is homotopic to idx rel{x}. If / : X ^- X is a map and r is a base path from x to f{x), define^ R(f, x, [r]) = d#(R(ufd, v, [u o r])). This extends the Reidemeister trace to self maps of finitely dominated spaces, in particular to compact ANR's. I must show it is well-defined. For this I need the Commutativity Lemma 2.3 below; it appropriately expresses the naive idea trace(A5) = trace(B A). Let (K, v) and (K\ v^) be finite path connected pointed CW r

complexes, let K^

K' be maps, let v be a base path in K^ from v^ to r(v) and let /x be a s

base path in K from v to s(v^). Write G = 7t\ (K, v) and G' = 7t\ (K\ v^). Let 0 = 5"# o r# and xj/ = r#os#. Then r#: ZG^ -^ "^^f is well-defined.^ LEMMA 2.3 (Commutativity). r#(R(s o r, i;, [/x(^ o y)])) = R(r o s, v\ [v(r o /x)]). PROOF. With the indicated base paths, sor = sr and r o 5 = r^. In the spirit of Footnote 5, the lemma boils down to noting that r#(q(trace Rs#(S))) = q\iv2icer#(R)r#s#{S)) which equals q\tmce(Sr#(R))) since r#os# = \l/. Here, for each k, R is the ZG' matrix representing rk : Ck{K) -> Ck{K'), and S is the ZG matrix representing Sk '• Ck{K') -^ Ck(K). D

PROPOSITION 2.4. R(f, x, [r]) is well-defined. d' PROOF.

Let ^^ ^ X be another finite domination. By 2.2 and 2.3 u'

R(ufd, f, [u o r]) = R{ud'u'fd'u d, f, [uod' ou' o r]) = (ud')#{R{udud'u'fd\ v\ W o r])). Applying d# to both sides gives the result.

D

It follows that Proposition 2.1 and Corollary 2.2 hold for the extended definition; moreover if h:X ^^ X' is a. homotopy equivalence between finitely dominated spaces and if f :X -^ X and f \X' ^^ X' are maps such that ho f is homotopic to f^ oh then h# maps the Reidemeister trace of g to that of /^; I leave the precise statement to the reader. When X is path connected and finitely dominated and / : X ^^ X is a map, define A^(/) to be the number of non-zero coefficients in the element of ZG^ which is Rif,x, [r]). ^ If the following diagram of groups and homomorphisms commutes G

^G

G'

^G'

4 ^ i^ then there is an induced function x# • G^ -^ G',, hence an induced homomorphism x# * ^ G ^ -> ZGj, .

508

R. Geoghegan

And define L(f) to be the sum of all the coefficients in R(f,x, [r]). There is, of course, a competing definition of L ( / ) , namely L\f) = Ylk^o(-^)^ ^^^^^(fk • Hk(X) -^ Hk(X)), but: PROPOSITION 2.5.

L(f)

=

L\f).

d

PROOF.

With KT^XSL finite domination as before, L\f)

= L\f

o d ou) = L' (u o f o d).

u

And I have already remarked that L ( / ) = L{uofod)

= L'{uofod).

D

NOTE. In my paper [13], where I present the material in this section, I attribute all this to Reidemeister [43] and Wecken [50], even though they wrote too early to have known about CW complexes and cellular chains. I learned it by reading their papers and adding what, by 1980, was common knowledge among topologists. Apart from the extension to finitely dominated spaces, I did not feel I was doing anything original. A somewhat different view is expressed in [22] which appeared at about the same time as [13] and includes this material. The term "Reidemeister trace" translates the German term used by Wecken [50]: in [22] it is given the name "generalized Lefschetz number". The latter term means something different in [12]. I urge a return to the original name "Reidemeister trace".

3. Equivalence of the two theories on ENR's ENR's are topological retracts of compact polyhedra, while finitely dominated spaces are homotopy retracts of compact polyhedra. I will show that the definitions in Sections 1 and 2 agree where both make sense, on ENR's. First, I relate fp classes to semiconjugacy classes in the general setting of a map f :X -^ X on a path connected, locally connected, semi-locally simply connected space. As before, write G = Tt\ (X, v) and let p: (X, v) -> (X, v) be the universal cover. Let x e Fix(/) and let /x be a path from f to x. Let x = il(l), where /x(0) = v. Then f(x) = xg for some g e G (again the usual left action of G on X is turned into a right action). Here is an exercise in covering space theory: PROPOSITION

3.1. g = [/x(/o/x)"^r~^] e G.

Thus the semiconjugacy class which (p associates with the fp class of x can also be found by looking at how / acts on any element of the fiber of p over x. Next, consider a special case. Let / triangulate a compact ^-manifold in R'^, let A^ be a subdivision of 7, and let /^: ^ ^- 7 be a simplicial map with finite fixed-point set, each fixed point lying in the interior of an n-simplex of K\ this implies there can be at most one fixed point in each a^. Write f: K ^^ K for the map /^ considered as a cellular map onK. 3.2. Let x e Fix(/) n a". Define e = 0 [resp. 1] when f is orientation preserving [resp. orientation reversing^ near x. Then L(f, x) = (—1)"^^. PROPOSITION

Nielsenfixedpoint theory

509

Without loss of generality, assume that x = 0 G M'^ and that on a" / agrees with a linear isomorphism A : R" ^- R'^ for which 1 is not an eigenvalue, and which maps a" onto an «-simplex r in R" (a simplex of 7) where a dr. Then A has no eigenvalue in the open interval (0,1), for otherwise A would not map da onto 9r. Thus there are no eigenvalues in [0,1], so sign(det(/ - A)) = sign(det(-A)) = (—1)"+^. D

PROOF.

Combining Propositions 3.1 and 3.2 yields (for this special case): PROPOSITION 3.3. The respective definitions of R(f, v, [r]), N(f) Sections 1 and 2 agree.

and L(f) given in

By well-known methods of transversality, in particular by a version devised for this purpose called the Hopf Construction (see [2, p. 117]), any map 171 ^- 171 is homotopic to a map resembling f with respect to a suitable K. This, together with the Commutativity and Homotopy Invariance properties stated in Sections 1 and 2, proves: 3.4. For any map f :Z ^^ Z on a compact ENR the respective definitions of /?(/, i;,[r]), A^(/) and L(f) given in Sections 1 and 2 agree.

THEOREM

Indeed, this theorem remains true if ENR is replaced by ANR, where the geometric definitions for ANRs are given in [2, Chapter 5].

4. Reducing the number of fixed points Let M be a compact piecewise linear (PL) ^-manifold in R" and let / : M ^- M be a map.^ Then / has finitely many fp classes, A^(/) of which have non-zero fp index. Intuition suggests it ought to be possible to find g homotopic to / such that g has exactly A^(/) = N(g) fixed points; the idea being that whole fp classes should coalesce to isolated fixed points and that those isolated fixed points of index 0 should be removable. This intuition turns out to be correct when n ^2 but false when n = 2. The case n = I is elementary [2, p. 107]. The method described in this section works for all n > 3; see for example [2, Chapter 8] or [6]. However, to shorten the proof I will assume n^4. I need a special homotopy extension lemma. For 5 c X, a homotopy H: B x I ^^ X is special if Fix(Ht) is the same for all / G /. If a map f :X ^^ X satisfies f{b) = H(b, 0) for all Z? G B, the resulting map (X x {0}) U (5 x /) ^- Z is a special partial homotopy. The proof of the following elementary lemma is given in [27, § 2]: LEMMA 4.1. IfX and B are ANR's with B closed in X, every special partial homotopy as above extends to a special homotopy X x I ^^ X. ^ The restriction to n-manifolds in R" allows us to give the main ideas with a minimum of technicalities; this case includes regular neighborhoods of polyhedra in R". For an arbitrary «-manifold a few technical adjustments are needed which are made explicit in [6, § 11]. For the polyhedral case see [2, Chapter 8].

510

R. Geoghegan

Returning to my set-up, I will assume that a preliminary homotopy has been performed so that / is PL with finitely many fixed points. Consider two of these fixed points, u and w, in the same fp class. Pick an embedded PL arc A in M joining u and w such that A n Fix(/) = 0, and the paths A'^^ M and / | A : A -> M are homotopic rel 9 A. Choose a PL map F:(Mx {0}) U (A x /) ^ M with F(x, 0) = f(x) for all JC G M, F(a, t) = f(a) when a = u or w, and F(a, 1) = a for all a e A. By general position (since n^4) assume F(A X (0,1)) n F(d(A X /)) = 0. Thus F\(M x {0}) U (A x [0,1 - 8]) ^ M is a special partial homotopy for any 8 >0. Let A/^ be a regular neighborhood of A inM withFix(/)nA^ = {u, w}.LQte > 0 be such that N2s(A) C N and NjsiM) retracts to M; here, Na{B) denotes the a-neighborhood of B in W. Pick 0 < 5 < 1 so that J(Fi_6,idA) < e. Setting F{xj) = f(x) when X G cl(M — A^) gives a special partial homotopy F : M x {0} U ((cl(M — N) U A) x [0, 1 — 5] -> M extending the appropriate part of F. By Lemma 4.1, this extends to a special homotopy F: M x [0,1 — 8] ^^ M. Let g = F\-s:M -^ M. Pick a regular neighborhood A^' of A such that A^^ c A^^CA) 2indd(g\N\ idyy/) < s. Then (id-g)\: A^' - {u, v} -^ A^^(0)-{0} C R" -{0}. It follows that the degree of ( i d - ^ ) I : dN' -^ W-{0} is i(g, M) + t(g, w) = L(f, u) -\- L(f, If), because the homotopy is special and degree is a homotopy invariant.^ There are two cases. If L(/, u) -h^(/, lu) = 0 then (id — g)\dN^ extends to a map k\N' ^^ Ne (0) - {0}. Define h\M^Why:h = g outside A^' and /i = id - )^ on N'. Then h is PL, Fix(/i) n A^^ = 0, Fix(/i) = Fix(/) - {M, U;}, /Z(M) C M (because N28{A) C M) and h is homotopic in M to g (because N2s(M) retracts to M). In this way I "cancel" fixed points of opposite fp index in the same fp class. The other case, i(f, w) -\- i{f,w) =d ^ 0, is handled similarly. One thinks of A^^ as a cone^ with base dN' and vertex u. Define k: (A^', A^' - [u}) -^ (Ns(0), Ne(0) - {0}) by coning the map (id — g)\dN\ and write h = id — k on N^ and h = g outside A^^ Then h : M ^^ M cM.^, h is homotopic in M to g, Fix(h) = Fix(/) — {w;}, and i{h,u) = d. In summary, I have proved a special case of the main theorem on reducing the number of fixed points of maps on manifolds: 4.2. Let f: M ^^ M be a map on a compact topological n-manifold, n ^2. Then f is homotopic to a map with precisely N {f) fixed points.

THEOREM

I have already discussed the failure of this theorem for ^z = 2 in the Introduction. NOTE. Theorem 4.2 was proved for PL manifolds by G.H. Shi developing a less general version due to Wecken [50]. In fact Shi's result also holds for compact polyhedra of dimension ^ 3 such that all links of vertices are connected. The extension to topological manifolds was given by R.F Brown. See [2, Chapter 8] for details and references. Theorem 4.2 was extended to compact polyhedra of dimension < 2 such that all links of vertices are connected, except for surfaces, in [27]. ^ In filling in the details one must use the fact that a regular neighborhood of a ball is a ball, of a triad of balls is a triad of balls, etc. See [6].

Nielsen fixed point theory

511

Part II. Some connections with other areas 5. The Euler characteristic and a problem about traces of idempotents When X is finitely dominated the Euler characteristic of X, x(^)» is defined to be Hk^oi-^f r2ir\kHk{X). By Proposition 2.5, x ( ^ ) = ^(idx). But what is /?(idx, x)? For guidance, consider the special case of a compact path connected ENR, Z. Every point z G Z is a fixed point of idz and all points are in the same fixed point class. Combined with Proposition 2.5 this gives: 5.1. When Z is a path connected compact ENR, R(idz, z) = x ( ^ ) [ l ] ^ Gi where [I] E G\ is the trivial conjugacy class. Hence N{idz) =^ or \ depending on whether x (Z) =0 or x (Z) / 0. PROPOSITION

Proposition 5.1 suggests what should hold for any finitely dominated complex X: CONJECTURE 5.2. R(idx,x)

= x(X)[l]. Equivalently, Niidx) = 0 or 1.

This turns out to be equivalent to a well-known conjecture in algebraic A^-theory called the Strong Conjecture of Bass [1, p. 156]. Let A be an idempotent ZG-matrix and as before let q : ZG -> ZG i be induced by the function G ^- G i sending each ^ G G to its conjugacy class. STRONG BASS CONJECTURE.

^(trace(A)) is an integer multiple of the trivial conjugacy

class. The word "strong" is explained in the note at the end of this section. To explain the point, I must recall the Wall finiteness obstruction and related matters. Denote the isomorphism class of a finitely generated projective ZG-module P by [P]. The Abelian group Ko(ZG) is defined to be the free Abelian group generated by all such [P] modulo the relations [P] -\- [Q] = [P ^ Q]. Let {P}bc the corresponding element of KQ{7JG). There is a natural monomorphism Z -^ Ko(ZG) taking 1 to {ZG}. The cokemel, Ko(1iG), is the reduced projective class group of the ring ZG. We write n : KQ{7JG) -^ Ko(ZG) for the quotient homomorphism. If A is an idempotent m x m ZG-matrix, the image of A: ZG^ -^ ZG^ is a projective ZG-module; conversely if P is a finitely generated projective ZG-module it is a direct summand of a free module ZG'^, and the projection of ZG'^ onto the submodule P of ZG^ is represented by an idempotent matrix. So elements of Ko(ZG) can be regarded as equivalence classes {A} of idempotent ZG matrices A. The homomorphism A i-> trace(A) h^ ^(trace(A)) well-defines the^ K-theoretic trace To:Ko(ZG) -^ ZG\. One thinks of TQ{{A}) as a sort of "rank" of the projective module represented by A, and ^ The Abelian group IJG\ is canonically isomorphic to HHQ(ZG), where HH^^iZG) is the Hochschild homology of the ring ZG. This ^-theoretic trace is variously known as the Hattori-Stailings trace or the Dennis trace.

512

R. Geoghegan

the Bass Conjecture says that this rank is essentially an integer and not a complicated element of ZG \. If X is a finitely dominated space whose fundamental group is G there is an element (7(X) G Ko(ZG), called the Wall finiteness obstruction, which is zero if and only if X is homotopy equivalent to a finite complex; see [21, Theorem 6.8] for a discussion. Choose A so that n({A}) = a(X), and consider 7b({A}) e G\.lf I is SL k x k identity matrix then 7t({A}) = n({A e /}) and To({A 0 /}) = To({A}) + k[ll so the sum of the coefficients in To({A}) is not determined by (J(X), but can be any integer. However, if n({A}) = 7T({B}) then To (A) — To{B) can only be non-zero in the component of the trivial conjugacy class [1] G Gi. The following is essentially proved in [13]: 5.3. If7T({A}) = cr(X) and A is chosen so that the sum of the coefficients in 7b({A}) is x(X) then Riidx,x) = ro({A}) = ^(trace(A)).

THEOREM

If Conjecture 5.2 holds for X then ^(trace(A)) = x(^)[l]» so the Strong Bass Conjecture holds. Conversely, if ^(trace(A)) = n[l] for some integer n (as asserted by the Strong Bass Conjecture) then we have chosen A so that n = x ( ^ ) . implying that Conjecture 5.2 holds for X. So Conjecture 5.2 is true for X if and only if the Strong Bass Conjecture holds for all idempotent matrices A such that 7r({A}) = cr(X). In particular, if cf(X) = 0, comparing the matrix A of Theorem 5.3 with the idempotent matrix 0, we have: COROLLARY

5.4. If X has the homotopy type of a finite complex then Conjecture 5.2

holds for X. Note that, by [51], Corollary 5.4 includes the case of X a compact ANR, and in particular a compact topological manifold. The Strong Bass Conjecture is known to hold for all idempotent ZG matrices when G is residually finite, or when G is of type F (i.e., there exists a finite K{G, 1) complex) and is either a subgroup of GL^ (C) or has rational cohomological dimension ^ 2, so Conjecture 5.2 is true for such fundamental groups; see [1,9,34,44]. I regard Conjecture 5.2 as one of the deepest unsolved problems of Nielsen Fixed Point Theory. The algebraic problem posed precisely by Bass was recognized, if vaguely, in [49, §4]. The Weak Bass Conjecture says that the sum of the coefficients of non-trivial conjugacy classes in ^(trace(A)) is 0; a corresponding weaker version of Conjecture 5.2 can easily be stated. The Weak Bass Conjecture was first stated in [8]. Note that Conjecture 5.2 is mis-stated in [28]. NOTE.

6. Gottlieb's theorem on the center of a group An elegant application of fixed point theory to group theory was pointed out by Gottlieb [19]. It has been used many times by geometric topologists. The key lemma is:

Nielsen fixed point theory

513

LEMMA 6.1. Let K be a finite connected CW complex, v a vertex of K, and F: K x I -^ K a homotopy with FQ = Fi = idK- Write G = 7ti(K, v) and let the loop F(v, •) represent g E G. If the Euler characteristic x (K) ¥" ^ ^^^^ ^ = 1PROOF. Apply Corollary 2.2. In this case R(Fo, v) and R(F\, v) both lie in Gi, and one has x(X)[l] = x(X)[l]g = x(X)[gl hence ^ = 1. D

The elements g e G which arise, as in Lemma 6.1, from a homotopy id A: :^ id A: form a subgroup, G(K), of G called the Gottlieb subgroup. Lemma 6.1 says that when X (K) ^ 0, g(K) = {!}. Obviously g(K) is a subgroup of Z(G), the center of G. When K is aspherical GiK) = Z{G), as can be seen by elementary obstruction theory: build the homotopy F :K X I ^^ K for any given g e Z(G), skeleton by skeleton. Recall that a group G has type F if there is a finite ^ ( G , 1) complex K. For such a group the Euler characteristic, x (^)^ is defined to be x ( ^ ) . The above remarks and Lemma 6.1 imply: THEOREM

6.2 (Gottlieb's theorem). IfG has type F and x(G) ^ 0 then G has trivial

center. Subsequently, using different methods, Eckmann and Rosset, [43], enlarged the conclusion of Theorem 6.2 to: .. .then G has no non-trivial Abelian normal subgroup. I have stated Gottlieb's theorem in the traditional way, but there is a theoretical case for stating a (possibly) more general version. A group G has type FD if there is a finitely dominated K{G, 1) complex. As I write, it is unknown whether or not "type FZ)" implies "type F". The algebraic definition of /?(/, v) was extended to finitely dominated spaces in Section 2. Using the equivalence of Conjecture 5.2 and the Strong Bass Conjecture as explained in Section 5, one easily adapts the proofs of Lemma 6.1 and Theorem 6.2 to get: THEOREM 6.3. IfG has type FD, if the Strong Bass Conjecture holds for all idempotent XG-matrices, and ifx(G) ^ 0, then G has trivial center.

In the context of one-parameter fixed point theory, specifically in [16], a "one-parameter Euler characteristic", x\ {G), is defined for groups G of type F (in that context one thinks of X (G) as "0-parameter") and the following is proved: THEOREM 6.4. If G has type F, if the Strong Bass Conjecture holds for all idempotent XG-matrices, and /fxi (^5 Q) 7^ 0, then G has infinite cyclic center.

The definition of xi cannot be given here, but Theorem 6.4 is a "one-parameter Gottlieb theorem". The apparent need for the hypothesis about the Strong Bass Conjecture is connected with the need^^ for it in Theorem 6.3. I end this section by pointing out a useful generalization of the Gottlieb subgroup. Let / : {K, v) -> {K, v) be a cellular pointed map on a finite CW complex, inducing (l):G ^ G as in Section 2. Let F : ^ x / ^ - ^ b e a homotopy with FQ = F\ = / , and let g G G be ^^ For perfect symmetry, the hypothesis of Theorem 6.4 should be weakened from "type F " to "type FD'\ This can probably be done.

514

R. Geoghegan

represented by the loop F(i;, •); as F varies among possible homotopies, the elements g which arise this way form a subgroup T ( / , v) of G called the Jiang subgroup. In particular, T(idK, v) = G(K). Here is a useful theorem. 6.5. Let T(f, v) = G. Then 0(G) lies in the center Z(G)\ for some L e Z, R(f, v) = i. Y^ceG^, ^ ' L(f) = i.N(f); L ( / ) = 0 if and only ifN{f) = 0 if and only if Rif v) = 0; when N(f) > 0 (equivalently, i ^ 0) the injection 0 from the set offp classes off to the set G(p of semiconjugacy classes is a bijection (hence G(j) is finite and all semiconjugacy classes occur as Jp classes).

THEOREM

(i) (ii) (iii) (iv) (v)

PROOF. Clearly T ( / , v) commutes with (/)(G), and (i) follows. When R(f v) = 0, (ii) is trivial; otherwise (ii) and (v) follow from Corollary 2.2 because PgR(f, v) = R(f, v) for every g eG. Parts (iii) and (iv) follow from the definition of L ( / ) and A^(/) in Sections 1 or 2. D

The hypothesis T(f, v) = G holds in various important cases: examples are given in [2, Chapter?]. I have given Gottlieb's original proof of Theorem 6.2. Better known is the proof in terms of finitely generated free ZG-resolutions given by Stallings [49]. I like Gottlieb's proof because Lemma 6.1 holds for all finite complexes, aspherical or not. In [16], Theorem 6.4 is proved assuming only the Weak Bass Conjecture over Q, but this requires a more sophisticated replacement for Theorem 6.3. NOTE.

7. Periodic points, zeta functions and torsion A. Periodic point theory Let / : (X, v) -^ (X, v) be a (pointed) map, let G = 7T\ (X, v) and let / induce (t):G -^ G. Throughout this section Z is a compact ENR or a finite CW complex as appropriate (see Sections 1, 2), and when Z is a finite complex f is a vertex and / is a cellular map. A periodic point of period m is a fixed point of the m-fold iterate / ^ : X ^- X. Nielsen Periodic Point Theory is concerned with the question: what are necessary and sufficient conditions for f to be homotopic to a map g having no periodic points! Of course, just as in Section 4, one would only hope for a complete answer when X is a compact n-manifold, n^2,or when X is a suitable compact polyhedron as in [2, Chapter 8]. Whereas a satisfactory answer was given to the analogous question for fixed points in Part I, a complete answer to the periodic point question is not known as I write. A necessary condition for having such a g is that the sequence (Rif^)) e Y[^=\ ^G(^m be trivial. Hence, a necessary condition is that the sequence of Lefschetz numbers {L{f"^)) e Y[^=\ ^ ^^ trivial. For less crude but still computable sequences between these two extremes, one can consider a linear representation p : G ^- GLr(5') where 5 is a

Nielsen fixed point theory

515

commutative ring (with 1) and p ocj) = p. The corresponding sequence is {L(f^, p)) e n ^ = i Sy where L(f, p), the p-twisted Lefschetz number of / , is defined to be the image of /?(/, v) under the homomorphism induced by p. In detail, p induces p\ZG -> Mr(S), the ring of r x r matrices over S, and hence p* = trace o jo'.ZG -> S. Define L(f, p) = Y^j^yQ(—l)^ tmcQp^(fk) e S, where p* maps ZG-matrices to ^-matrices entry by entry. Of course, for this to make sense I assume here that / is a cellular map on a finite complex. From the ENR point of view the homomorphism p induces a function G^ -^ GLr(S) because p o 0 = p, so L(/, p) coalesces the components of R(f^ v) = ^ . classes F ^(/' F)0(F) to give first an 5-matrix, and then an element of 5 - its trace. When 5 = Z, r = 1 and p(G) = {1} C Z, then L(f, p) = L ( / ) . All this can be summarized, if tautologically, as: THEOREM 7.1. Let f: (X, v) -> (X, v) be a cellular map on a finite complex. If the sequence {R(f^, v)) is non-trivial, or ififor any linear representation pofG = 7T\ (Z, v) as above, the sequence {L{f^, p)) is non-trivial, then f is not homotopic to a periodic point free map. A more sophisticated view relates the sequences {L{f^, p)) to torsion and to zeta functions. Let TLG^iit]] denote the (p-twistedpower series ring over ZG, consisting of power series J2m>o ^mt^ in a formal variable t, with Um € ZG, subject to the multiplication rule tg = (t){g)t where g e G. When A is a square ZG-matrix, the matrix I — At is invertible in the ring of ZG^[[?]]-matrices; its inverse is X!m>o('^^)'^ • ^^^^ simple observation turns out to underlie everything. Recall the definition of K{ of a ring. For any ring S (with 1), GL(5) denotes the direct limit of GLi (5) ^-^ GL2(S) ^-> • • • where the rth inclusion takes the r x r matrix M to the (r + 1) X (r + 1) matrix M 0 [1]. The Abelianization GL(S)/[GL(S): GL(S)] is denoted by K\ (S). The torsion of M is the element of K\ (S) which M represents. A good reference for background is [4]. When Z is a finite complex and / is cellular, let A(f v) e ^i(ZG*^[[f]]) be represented by the matrix c(f) = n^>o(^ — fkt)^~^^ • While this matrix depends on choices (see Section 2), Z\(/, v) does not [15, Proposition 5.3]. If p : G -> Gl^riS) is as above, p induces a homomorphism K\{IjG^[[t]]) -^ K\{S[[t]Y). The image of A{f, v) is Z\(/, p) represented by c(/, p) = O^^o^^ ~ P*(^)0^~^^ • Applying the determinant map det: Kx (5[[/]]) -> S[[t]] we get detZ\(/,p)=f]det(/-p*(/^)r)^

^^'^'eSmi

Define the p-twisted zeta function, f (/, p) e S[[t]], to be exp(X;^^i ^L{f'^,p)t'^). "rationality of the zeta function" is expressed by: THEOREM

Then

7.2. detz\(/,p) = f ( / , p ) .

I will discuss the proof of this theorem later in the section. For now, the point to note is that {L{f^, p)) is trivial if and only if ^ ( / , p) is identically 1. So we have:

516

R. Geoghegan

COROLLARY 7.3. With hypotheses as in Theorem l.l, ifdetA(f, p) is not identically 1 then f is not homotopic to a periodic point free map. REMARK. Z \ ( / , i;) is the torsion of the matrix c ( / ) , and A{f, p) is the torsion of the matrix c(f, p). For historical reasons the torsion of detc(/,p) is called "Reidemeister torsion". So the zeta function ^ ( / , p) is a Reidemeister torsion element.

All this is usually discussed in the context of the suspension (semi)flow on the mapping torus of / , as I now explain. B. Dynamics If F is an fp class of the map f^, so are / ( F ) , . . . , /^~^ (F), and the corresponding mperiodic orbit class of / is U7=o / ^ (F)' ^^^^^ /^(^) = ^ Note that the fp classes / ' (F) and /^(F) are either identical or disjoint. One classifies m-periodic orbit classes algebraically thus: the relation g ^ (l)(g) well-defines an equivalence relation on the set G^m; write G(f)m/{(/)) for the set of equivalence classes. By analogy with 0 in Section 1, there is an injective function 0^ from the set of m-periodic orbit classes of / to the set G(pm/{(l)). When X is a compact ENR, define

m-periodic orbit classes O

Under the obvious quotient homomorphism R(f^, v) is taken to R(f^, v), the effect being to coalesce the distinct fp classes of / ^ which are in the same m-periodic orbit class of / , adding up the fp indices of the distinct pieces. Let T(f) be the mapping torus of / , i.e., the quotient space X x //(x, 1) ^ (f(x), 0). Its fundamental group ^^ is F = {G,t \tgt~^ = 0 (g) Vg G G); when 0 is an isomorphism this is the semidirect product of G and Z over (p. As before, Fi denotes the set of conjugacy classes in F , and the function G^w -^ Fi sending the semiconjugacy class of g to the conjugacy class of gt^ maps UmeZ^^"' surjectively onto F\, and induces a bijection L[mez(^0'" /<^)) -^ ^1 • Write ^ : 0 ^ Z[G0A^V(0>] -^ ZF\ for the induced isomorphism, and write ^ ( / ^ , v) = ^(Rif^, v)). Thus,

% " ' ^) =

L

i{r.O)^0\O) e Zr,;

m-periodic orbit classes O

R{f^^ ^) and R{f^, v) encode exactly the same information. The algebraic formula for R(f^, v) is worth noting. Abusing notation, let q :ZF -^ ZF\ be the quotient homomorphism induced by the function F ^^ F\ sending each element to its conjugacy class. ' ' The use of t to be an element of F here and a formal variable elsewhere is deliberate and is intended to be suggestive. See §5 of [15].

Nielsen fixed point theory PROPOSITION PROOF.

ir)k

517

7.4. Rif"^, v) = J]^^o(-l)^^(trace(/^0'")= fk • Hfu) • .•(t>'^-\fk), so in Z r one has {r)kt'^

= (fktr.

D

There is a suspension semiflow on r ( / ) , i.e., a continuous action of the additive semigroup of non-negative real numbers, in which each point flows "forward" round and round the mapping torus. In the most interesting case, / is a diffeomorphism of a compact manifold, and then the semiflow is smooth and extends to a suspension flow (a smooth action of R). A point of period m corresponds to a time m closed orbit in the semiflow. One easily proves: PROPOSITION 7.5. Points y and z ofperiod m give rise to freely homotopic time m closed orbits in the suspension semiflow if and only if y and z are in the same m-periodic orbit class of f.

This is why m-periodic orbit classes are useful. The Lefschetz-Nielsen series of / is the sequence {R(f^^v)) ^ Y\m>\ ^[G^^'"/{0>]. or, equivalently, {R{f^, v)) e Y\m>\ ^ A - There is a corresponding sequence of non-negative integers {N(f^)) where N (f^),thQ Nielsen number of time m closed orbits, is the number of non-zero components in R{f^, v) or R(f^, v). By Proposition 7.5 we have: PROPOSITION 7.6. There are at least N{f^) time m closed orbits in the suspension semiflow associated with / .

For p:G ^^ GLr(S) as above, the induced homomorphism which takes R{f^, v) to ^(/'"^ P) factors through Z[G0m/(0)], so the sequence (L(f^, p)) is also a "linearization" of the Lefschetz-Nielsen series. In fact, there is a homomorphism

Kx{ZG^[[t]-\) -^ W Z[G^^/((/>>] ^ 5[M] givenexplicitly in the proof of Theorem 5.13 of [15] which maps Z\(/, v) i-> {R{f^, v)) \-^ ^ ( / , p). Theorem 7.2 follows from the fact that this homomorphism equals

A{f,v)^A{f,p)^deiA{f,p). In fact, Theorem 7.2 is a corollary of a deeper "rationality theorem" [15, Theorems 5.6 and 5.10] which cannot be stated at the level of this article. ADDED IN PROOF. This subject has seen important development since this chapter was written. See, for example, [23,24,39-42,45,46,48]. D NOTE. The contents of this section are distilled from Sections 5, 6 of [15], somewhat influenced by [31]. The Lefschetz zeta function appeared in [37] and various zeta functions off(/,p)-typein[12].

518

R. Geoghegan

8. Growth rate of Nielsen numbers The growth rate of a sequence {am) of complex numbers is growthfl^ = m a x | l,limsup|a^|^/^ I; the value oo is permitted. When growth a^ > 1, {am) grows exponentially. I will describe some results about the growth rate of numbers associated with Z'" : X ^- Z as m -^ oo. For any set C, the free Abehan group ZC is given the L^-norm: || X^c^c'n(c)c||zc = 5^J^2(c)|; the subscript ZC will often be omitted. \\R{f'^,v)\\ = i^fixedpoim k ( / ' " , F ) | classes F

and \\R{f^,v)\\

= XI "'-orbit | K / ^ , 0 ) | . Recall from previous sections the numbers classes O

Nif^) and Nif"")- Asymptotic versions of these are A^^(/) = growth^_^^A^(/^) = growth^^^ N(D and R^(f) ^ growth^^^ \\Rif^, v)\\ = growth^^^ |hR(/-, i;)||. The equalities are consequences of the inequalities N{f^) ^ N{f^) < mN{f^) and

mr, v)\\^ \\R{r^ v)\\ ^mmr^ VM

If C is a ring the definition of norm extends to ZC matrices. For such a matrix M define \\M\\= Y.i j Wijl Then ||MA^|| ^ ||M|| ||A^|| when M and A^ can be multiplied, and II trace Mil ^ ||M||. PROPOSITION 8.1. N"^(f) PROOF.

^ R^(f)


The first inequality is clear. For the second,

\R{r

.V

=

^(-l)*^(trace((r),)

<Elk(t--((r).)II.G,.<EI t r a c e l l r ) , ) ! ZG k

< EII(/'")JIZG k

l^ •

k

-'(/OllzG

< / ) " ' •

Y^ll r \\m k

So R'^if)

< maxgrowth(|| A f ) < oo.

^

m^-OQ

The next result illustrates how classical complex analysis enters, when the zeta function has a complex rather than a formal variable. PROPOSITION 8.2 (Lower bound for R'^if)). Let p:G -> GU(C) be a representation satisfying p ocp = p. Let [im ^ sup{| trace p 0 ( F ) | | F is anfp class of / ^ } , and let JJL = growth^^^ Ijim- For any zero or pole w of the rational function f (/, p) e C{t), R^(f) ^ —r^. In particular, if p is a unitary representation, R'^(f) ^ y^.

Nielsen fixed point theory

519

PROOF, f (/, p)(0) = 1, SO u; ^ 0. We have log f (/, p){t) = Y.m^x ^R(r,p)r. Now, growth L(f^,p) is the reciprocal of the radius of convergence of log ^(f,p), and is therefore ^ ^ . But R(r,v)

= E F K / " , F ) 0 ( F ) , SO \L(r,p)\

I trace(p0(F))| ^ ||/?(/^, i;)||/x^. So /x/?~(/) ^ ^ .

^ EFI^C/'^'F)!

D

8.3 (Upper bound for R^ (/)). Let Fk denote the matrix whose (i, j) entry is \\{fk)ij llzG- Then R^(f) ^ meiXk (spectral radius of Fk). PROPOSITION

PROOF. AS

in the proof of Proposition 8.1,

So R^if) < max^(growth^^^ trace((F^)'")) = max/:(spectral radius of Fk). I will end with a theorem relating asymptotic Nielsen theory to topological entropy. Let (X, d) be a compact metric space, and let / : Z ^- X be a map. For s > 0 and n ^ 1, a subset £" of X is (n, e)-separated under / if for each pair x ^ y in E there is 0 ^ i < n such that d(f^(x), f^iy)) > e. Let Snis, f) denote the largest cardinality of (n, £)-separated subsets E under / . Thus Sn (s, f) is the greatest number of orbit segments {x, / ( j c ) , . . . , /"~^ (jc)} of length n that can be distinguished one from another when we can only distinguish between points of X that are at least 6 apart. Now define /?(/,£:) =lim sup-log ^^(e,/)

and h(f) =

\imh(f,e).

The number 0 ^ h{f) ^ oo, which is easily seen to be independent of the metric d, is the topological entropy of / . If h{f, e) >0 then, up to resolution e > 0, the number Sn (s, f) of distinguishable orbit segments of length n grows exponentially with n. So h(f) measures the growth rate in n of the number of orbit segments of length n with arbitrarily fine resolution. The following is due to Ivanov [25]. The proof given here is taken from [31]: D 8.4. Let f : X -> X be a map on a compact polyhedron, and let h(f) denote the topological entropy of f. Then h(f) ^ \ogN^(f).

THEOREM

Let 5 > 0 be such that every loop in X of diameter < 28 is homotopically trivial in X. Let e > 0 be a smaller number such that d(f(x), f(y)) < 8 whenever d(x, y) < 2e. Let £•„ C X be a set consisting of one point from each essential fp class of f^. Thus \En\ = N(f^). By the definition of h(f), it suffices to show that En is (n, ^)-separated. Suppose not. Then there would be two fixed points of f^,x ^y e En, such that d(f^ (x), Tiy)) ^s forO^i
520

R. Geoghegan

NOTE. This section is adapted from [31] where a richer discussion with worked examples can be found.

References [1] H. Bass, Euler characteristics and characters of discrete groups. Invent. Math. 35 (1976), 155-196. [2] R.F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, Chicago (1971). [3] R.F. Brown, ed., Fixed Point Theory and its Applications, Proceedings of a Conference at Berkeley, CaHfomia, Contemp. Math., Vol. 72, Amer. Math. Soc, Providence (1988). [4] M.M. Cohen, A Course in Simple-Homotopy Theory, Springer, New York (1973). [5] D. Dimovski, One-parameter fixed point indices. Pacific J. Math. 164 (1994), 263-297. [6] D. Dimovski and R. Geoghegan, One-parameter fixed point theory. Forum Math. 2 (1990), 125-154. [7] E. Dyer, Some remarks on fixed point theory. General Topology and Modem Analysis, Academic Press, New York (1981). [8] E. Dyer and A.T. Vasquez, An invariant for finitely generated projectives over 7LG, J. Pure Appl. Algebra 7 (1976), 1M-1\'^. [9] B. Eckmann, Cyclic homology of groups and the Bass conjecture. Comment. Math. Helv. 61 (1986), 193202. [10] E. Fadell and G. Foumier, eds. Fixed Point Theory, Proceedings of a Conference at Sherbrook, Quebec, Canada, Lecture Notes in Math., Vol. 886, Springer, New York (1981). [11] M. Feckan, Nielsen fixed point theory and non-linear equations, J. Differential Equations 106 (1993), 312331. [12] D. Fried, Homological identities for closed orbits. Invent. Math. 71 (1983), 419^^2. [13] R. Geoghegan, Fixed points infinitely dominated compacta: the geometric meaning of a conjecture of H. Bass, Shape Theory and Geometric Topology, Lecture Notes in Math., Vol. 870, Springer, New York (1981), 6-22. [14] R. Geoghegan and A. Nicas, Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces, Amer. J. Math. 116 (1994), 397^46. [15] R. Geoghegan and A. Nicas, Trace and torsion in the theory of flows. Topology 33 (1994), 683-719. [16] R. Geoghegan and A. Nicas, Higher Euler characteristics (I), Enseign. Math. 41 (1995), 3-62. [17] R. Geoghegan and A. Nicas, A Hochschild homology Euler characteristic for circle actions, AT-Theory 18 (1999), 159-178. [18] R. Geoghegan, A. Nicas and J. Oprea, Higher Lefschetz traces and spherical Euler characteristics. Trans. Amer. Math. Soc. 348 (1996), 2039-2062. [19] D. Gottlieb, A Certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840-856. [20] M. Hirsch, Differential Topology, Springer, New York (1976). [21] B. Hughes and A. Ranicki, Ends of Complexes, Cambridge Tracts in Math., Vol. 123, Cambridge University Press, Cambridge (1996). [22] S.Y. Husseini, Generalized Lefschetz numbers. Trans. Amer. Math. Soc. 272 (1982), 247-274. [23] M. Hutchings and Y.-J. Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of three manifolds. Topology 38 (1999), 861-888. [24] M. Hutchings and Y.-J. Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), 369-396. [25] N.V. Ivanov, Entropy and the Nielsen numbers, Soviet Math. Dokl. 26 (1982), 63-66. [26] J. Jezierski, One-codimensional Wecken type theorems. Forum Math. 5 (1993), 421^39. [27] B.J. Jiang, On the least number of fixed points, Amer. J. Math. 102 (1980), 749-763. [28] B.J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math., Vol. 14, Amer. Math. Soc, Providence, RI (1983). [29] B.J. Jiang, Fixed points and braids I, Invent. Math. 75 (1984), 69-74. [30] B.J. Jiang, ed.. Topological Fixed Point Theory and Applications, Proceedings of a Conference at Tianjin, China, Lecture Notes in Math., Vol. 1411, Springer, New York (1989). [31] B.J. Jiang, Estimation of the number of periodic orbits. Pacific J. Math. 172 (1996), 151-185.

Nielsen fixed point theory

521

[32] M.R. Kelly, Minimizing the number of fixed points for self-maps of compact surfaces. Pacific J. Math. 126 (1987), 81-123. [33] T. Kiang, Theory of Fixed Point Classes, Springer, Beriin (1989). [34] P.A. Linnell, Decomposition of augmentation ideals and relation modules, Proc. London Math. Soc. (3) 47 (1983), 83-127. [35] C.K. McCord, ed., Nielsen Theory and Dynamical Systems, Contemp. Math., Vol. 152, Amer. Math. Soc, Providence (1993). [36] J. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottsville (1965). [37] J. Milnor, Infinite cycle coverings. Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston (1968), 115-133. [38] J. Nielsen, Untersuchungen zur Topologie des geschlossen zweiseitigen Flache, I, II, III, Acta Math. 50 (1927), 189-358; 53 (1929), 1-76; 58 (1932), 87-167. [39] A. Pajitnov, Simple homotopy type of the Novikov complex and Lefschetz zeta-functions of the gradient fiow, Russian Math. Surveys 54 (1999), 119-169. [40] A. Pajitnov, C^-generic properties of boundary operators in the Novikov complex, pseudoperiodic topology, Amer. Math. Soc. Transl. Ser. 2 197 (1999), 29-115. [41] A. Pajitnov, Closed orbits of gradient flows and logarithms of non-Abelian Witt vectors, to appear in KTheory. [42] A. Pajitnov and A. Ranicki, The Whitehead group of the Novikov ring, to appear in A'-Theory. [43] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936), 586-593. [44] J.A. Schaeffer, Relative cyclic homology and the Bass conjecture. Comment. Math. Helv. 67 (1992), 214225. [45] D. Schuetz, Gradient flows of closed 1-forms and their closed orbits, to appear in Forum Math. [46] D. Schuetz, One-parameter fixed point theory gradient flows of closed one-forms. Preprint. [47] S.P. Singh, ed.. Nonlinear Functional Analysis and its Applications, NATO-ASI Series C, Vol. 173, Kluwer Academic Publishers (1986). [48] E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966). [49] J. Stallings, Centerless groups - an algebraic formulation of Gottlieb's theorem. Topology 4 (1965), 129134. [50] F. Wecken, Fixpunktklassen, I, Math. Ann. 117 (1941), 659-671; II, Math. Ann. 118 (1942), 216-234; III, Math. Ann. 118 (1942), 544-577. [51] I.E. West, Mapping Hilbert cube manifolds to ANR's: a solution to a conjecture ofBorsuk, Ann. of Math. 106(1977), 1-18. [52] X. Zhang, The least number of fixed points can be arbitrarily larger than the Nielsen number, Acta Sci. Natur. Univ. Pekinensis (1986), 15-25.

This Page Intentionally Left Blank

CHAPTER 12

Mapping Class Groups^ Nikolai V. Ivanov Michigan State University, Department of Mathematics, Wells Hall, East Lansing, MI 48824-1027, USA E-mail: [email protected]

Contents 1. Introduction 2. Topology of surfaces 2.1. The dimension 2 2.2. Classification of surfaces 2.3. Circles and arcs on surfaces 2.4. Pants decompositions 2.5. Geometric structures on surfaces 2.6. Spaces of diffeomorphisms and embeddings: generalities 2.7. Spaces of diffeomorphisms and embeddings: surfaces 2.8. Gluing discs to a surface 2.9. The Dehn-Nielsen-Baer theorems 3. Complexes of curves 3.1. Definitions 3.2. Connectivity ofC{S) and the Morse-Cerf theory 3.3. The homotopy type of the complexes of curves 3.4. Hyperbolic properties 4. Dehn twists, generators, and relations 4.1. Dehn twists 4.2. The Dehn-Lickorish theorem 4.3. Finite presentations 5. Teichmiiller spaces 5.1. Definitions 5.2. Length functions and Fenchel-Nielsen 5.3. The topology of Teichmiiller spaces 5.4. Comers and the mapping class groups 5.5. Ideal triangulations 6. Cohomological properties 6.1. Cohomology of groups and the Poincare-Lefschetz duality 6.2. Classifying spaces and universal bundles 6.3. Mess subgroups *Supported in part by the NSF Grants DMS-9401284 and DMS-9704817. HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved 523

flows

525 526 527 527 528 530 531 533 535 537 540 541 542 542 546 547 548 548 556 565 570 570 571 572 573 575 576 577 581 582

524

N.VIvanov

6.4. Virtual cohomological dimension 6.5. Homology stability 6.6. The low-dimensional homology groups 6.7. Virtual Euler characteristic 6.8. Torsion in mapping class groups, torsion in their cohomology and related topics 7. Thurston's theory and its apphcations 7.1. Classification of mapping classes 7.2. Measuring against circles and its applications 7.3. Action of the mapping classes on Thurston's compactification of Teichmiiller space 7.4. Some apphcations 7.5. Commuting elements and Dehn twists 8. Automorphisms of complexes of curves 8.1. The theorem about automorphisms 8.2. Intersection number 1 property 8.3. The case of a sphere with holes 8.4. Complexes of curves and ideal triangulations 8.5. An application to subgroups of finite index 8.6. An apphcation to Teichmiiller spaces 9. Mapping class groups and arithmetic groups 9.1. Arithmetic groups 9.2. Abstract commensurators 9.3. The original approach to the non-arithmeticity 9.4. Rank of the mapping class groups 9.5. The Mostow-Margulis superrigidity Acknowledgement References

584 588 591 594 595 599 599 602 604 605 608 609 609 610 613 614 615 617 618 618 619 620 622 624 625 625

Mapping class groups

525

1. Introduction The mapping class group Mods of an orientable surface S is defined as the group of isotopy classes of orientation-preserving diffeomorphisms S ^^ S. In addition to being a central object of the topology of surfaces (cf. 2.1), these groups also play an important role in the theory of Teichmiiller spaces and in algebraic geometry, where they are known under the name Teichmiiller modular groups or simply modular groups. Our notations are derived from the latter terminology. There are several closely related groups, which also deserve the name of the mapping class groups (or Teichmiiller modular groups). First of all, one may consider the extended mapping class group Mod^ of S defined as the group of the isotopy classes of all diffeomorphisms S -^ S. Tht pure mapping class group PMod^ of S is defined as the group of isotopy classes of all orientation-preserving diffeomorphisms S ^^ S preserving setwise all boundary components of S. Finally, one may consider the group Ms of all (orientation-preserving) diffeomorphisms S -^ S fixed on the boundary dS, considered up to isotopies fixed on the boundary. If 95 ^ 0, then diffeomorphisms fixed on 95 are automatically orientation-preserving. If 95 = 0, then, of course, Ms = Mod5^. All these groups could be also defined as the 0-th homotopy groups of suitable diffeomorphisms groups of 5. For example, Mod^ = 7ro(Diff(5)), where Diff(5) is the group of all orientation-preserving diffeomorphisms of 5 considered with, for example, C^-topology (or any other reasonable topology). The study of the mapping class groups was initiated in the 1920-ies by Dehn [43-45] and Nielsen [191-193]. Although their work had some common themes (cf., for example, the Dehn-Nielsen-Baer theorems in 2.9), in general their approaches were fairly different. Dehn was interested in the properties of the mapping class group as a whole, addressing, for example, such questions as the existence of a finite set of generators. He developed and exploited an important tool for this purpose: the action of the mapping class group Mod^ on the collection of the isotopy classes of all circles on 5. He called this collection the arithmetic field of 5. On the contrary, Nielsen was mainly interested in understanding the fine structure of the individual elements of Mod^. His methods draw heavily on hyperboHc geometry (a tool also favored by Dehn). For quite a while, the work of both Dehn and Nielsen was apparently forgotten. The ideas of Dehn related to the arithmetic field of a surface found a natural continuation in the ideas of Harvey [86,87] about the complex of curves of a surface, which is nothing else but the arithmetic field made in a simplicial complex in a natural way (see 3.1). A closely related object was considered in an influential paper of Hatcher and Thurston [91]. The ideas of Nielsen were partially rediscovered, extended and brought to an essentially complete form by Thurston in his theory of surface diffeomorphisms [53,222] (see Section 7). Later on, his theory was applied also to the structure of the mapping class group (and not only to its individual elements). In the present notes we have adopted a point of view going back to that of Dehn. Compared to the other thread, going back to Nielsen, it is more elementary and allows us to reach more quickly some really deep results about mapping class groups. Everyone interested in the mapping class groups eventually should learn the other point of view also (especially Thurston's theory of surface diffeomorphisms). We hardly do justice to it in a chapter devoted to Thurston's theory and some of its applications. If one takes Dehn's terminology seriously, our exposition is slanted toward the arithmetic of our subject, which

526

N.V. Ivanov

seems to be only natural for an introduction. The analogy with the arithmetic groups (see 9.1 for a definition), briefly touched on in Section 9, fits nicely in this approach (for example, the main results of Section 6 are, in fact, analogues of some fundamental theorems about arithmetic groups), giving a deeper meaning to the word arithmetic in this context. Our exposition is centered around three topics: the Dehn-Lickorish theorem providing a finite set of generators of Mod^ for closed S\ Harer's theorem computing the so-called virtual cohomology dimension of Mod^; and the author's theorem describing all automorphisms of Harvey's complex of curves. The first two topics are presented with complete proofs and a detailed review of the prerequisites; for the third topic only an outline is given. The proofs we give for the theorems of Dehn-Lickorish and Harer contain some new ideas and may be interesting even for experts. The table of contents gives a good idea of the topics included in these notes. The limitations of time and space prevented us from giving a more detailed treatment of many included topics and from discussing other topics, equally or, may be, even more important. As we already mentioned, our treatment of Thurston's theory is hardly adequate (although we hope that it is still useful); the same applies to the aspects of the theory of the Teichmtiller spaces related to the mapping class groups. Nothing is said about relations with algebraic geometry and mathematical physics; we refer to the recent survey of Hain and Looijenga [72] for this. Two omitted topics are very closely related to the ones included. In order not to pass them in complete silence, we very briefly discuss them here. The first one is the study of the so-called Torelli group, the subgroup of Mod^ consisting of the isotopy classes of diffeomorphisms of S acting trivially on H\ (S, Z). The pioneering work in this field is due to Johnson. In [115] he proved that the Torelli group of S is finitely generated if the genus of S is at least 3 and S has no more than 1 boundary component, and in [116,117] he computed the first homology group of the TorelH group (under the same restrictions). See [114] for a survey of this work. It was continued, in particular, in the algebro-geometric work of Hain; see [70], and especially the remarkable paper [71], where Hain proved the existence of a finite presentation for a Lie algebraic analogue of the Torelli group. Note that the question about the existence of a finite presentation of the Torelli group itself remains open. The second topic is the theory and applications of the characteristic classes of surface bundles (which are nothing other than the elements of the cohomology group //*(Mod5)), constructed independently by Mumford [188], Morita [168,169] and Miller [164]. Morita [168,169] and Miller [164] proved the nontriviality of these classes, and Morita [168,169] found a spectacular application of this nontriviality: he proved that the canonical homomorphism Diff(5) -^ Mods admits no section. In his further papers (see, for example, [170-173,177-179]) Morita studied these characteristic classes and appUed them to the theory of the Torelli group (extending in some respects the above mentioned work of Johnson) and to the Casson invariant of homology 3-spheres and related problems of the topology of 3-manifolds. We refer to his own surveys [174-176] for a discussion of these results. 2. Topology of surfaces In this section we attempt to oudine the material which should be at the foundation of the theory of the mapping class groups. All this material is fairly classical, although

Mapping class groups

527

some results (for example, the ones about spaces of diffeomorphisms and embeddings) are not so widely known as they deserve. Our discussion is more broad than is needed for later applications, in order to get closer to a coherent picture; still, our treatment is far from being complete. The proofs in this section are not to be taken too seriously; mostly, they are only outlines or just indications of the ideas involved in the real proofs.

2.1. The dimension 2 In order to put the topology of surfaces in a perspective, let us start with a particular view of the topology of manifolds in general. The central problem of the topology of manifolds is the problem of classification of manifolds. It is usually considered to be more of less solved for at least simply connected manifolds of dimension ^ 5. Another major problem is the problem of isotopy classification of diffeomorphisms. Consideration of diffeomorphisms up to isotopy distinguishes topology from other subjects, especially from the theory of smooth dynamical systems, which concentrates on the classification of diffeomorphisms up to conjugacy. A satisfactory solution of the isotopy classification problem for diffeomorphisms is known only for some special classes of manifolds. The problem which naturally comes after the classification of diffeomorphisms is the problem of understanding of the homotopy type of the diffeomorphism group of a manifold. It is extremely difficult even for such manifolds as spheres. These problems actually form a natural series. Namely, the classification of manifolds, the isotopy classification of diffeomorphisms, and the computation of homotopy groups 7ti (Diff(M)) for / > 0, where Diff(M) is the group of all diffeomorphisms M ^^ M (with, say, C^-topology), form a series of problems, parallel and related to the study of functors ^ i , K2, and Ki for / > 2 of the algebraic ^-theory. We can add also to this row the problem of the existence of a smooth manifold within a given homotopy type and the functor A^oIn the dimension 2, the solution of the classification problem is well known and is, in fact, a part of any introductory course in topology. The homotopy groups TTI (Diff(M)) with / > 0 are known for all compact two-dimensional manifolds, and, moreover, most of them are equal to 0. We will discuss these results in 2.7. This leaves us in the dimension 2 with only one of the above problems - the problem of the classification of diffeomorphisms. Interpreted broadly, this problem naturally includes a search for an understanding of the mapping class groups 7ro(Diff(M)), which is the main focus of our survey. But, before we concentrate on 7ro(Diff(M)), we will review some other parts of the topology of surfaces, partly because they are needed to understand 7ro(Diff(M)) and partly for their own sake.

2.2. Classification of surfaces Throughout the paper, all surfaces are assumed to be orientable and, unless the contrary is obvious from the context, compact.

528

N.VIvanov

There is no doubt that the reader is familiar with the classification of surfaces. Let us recall, for the sake of completeness, that an orientable connected surface S is determined up to a diffeomorphism by its genus g and the number of the boundary components b. Following Riemann, the genus can be defined as the maximal number of disjoint circles on S such that these circles do not separate S, i.e., the complement in S of the union of these circles is connected. The genus g and the number of boundary components b are related to the Euler characteristic x(S) of S by the well known formula x(S) = 2 — 2g— b.A surface of genus g with b boundary components is also known as a sphere with g handles and b holes; the terminology is justified by a well known picture. The standard proof of this classification theorem, based on triangulations and cutting and pasting arguments, provides a cannonical model of a surface of genus g with b boundary components. Namely, such a surface can be obtained from a 4g-gon by a well known identification of sides and removing b disjoint discs from the interior. The identification of sides is described by the word a\b\a^^b^^.. .agbga~^b~^ and this leads to a presentation of7ri(5): 7r\(S) =

[aub\,,..,ag,bg,d\,...,db\ a\b\a~^ b^ .. .agbga~^b~^d\ ...db = l).

If ^ > 0, then one can eliminate one J/ from this presentation and conclude that TTI (S) is a free group. One can obtain the same result in a more topological manner, by observing (using the above 4g-gon model, for example) that S has a 1-dimensional CW-complex as a deformation retract. But considering n\ (S) for surfaces with boundary as just a free group is to a great extent misleading, because in this way we lose all information about the boundary. It is much better to keep the track of the boundary by introducing the so-called peripheral structure - the set of 2b conjugacy classes of elements of TTI (5) corresponding to loops going once round a boundary component. It is also useful to consider the oriented peripheral structure - the set of b conjugacy classes of elements of 7T\ (S) corresponding to loops going once round a boundary component in the direction prescribed by a fixed orientation of S (which induces an orientation of the boundary dS). Note that all generators in the above presentation can be obviously represented by embedded loops, i.e., by loops / : [0, 1] ^- S such that f(x) = f(y) only if x = y ov [x, y} = {0, 1} (of course /(O) = / ( I ) is the base point).

2.3. Circles and arcs on surfaces Circles and arcs on surfaces and the action of diffeomorphisms on them are some of the main tools (and objects) of the topology of surfaces. By circles we understand the embedded circles, i.e. submanifolds diffeomorphic to the standard circle. Similarly, by arcs we understand submanifolds diffeomorphic to the interval [0, 1]. Usually, one restricts attention to properly embedded arcs; an arc / on a surface S is called properly embedded (or simply proper) if 31 = I DdS and / is transverse to dS. It is convenient to single out circles and arcs considered to be trivial. We call a circle C on a surface S trivial if C is either contractible in 5, or is homotopic to some boundary

Mapping class groups

529

component of S. Similarly, we call an arc / trivial if it can be deformed into the boundary of S in such a way that its boundary 9/ stays in dS. Also, it is important to distinguish between separating and nonseparating circles. We call a circle C on 5 nonseparating if 5 \ C is connected, and separating otherwise. As we noticed in 2.2 all generators in the standard presentation of 7T\{S) can be represented by embedded loops. The image of an embedded loop is a circle, and we call an embedded loop separating or nonseparating if this image is separating, or, respectively, nonseparating. The obvious loops representing a\,b\,.. .,ag,bg are nonseparating, while the obvious loops representing d\, ...,db are separating (in fact, their images can be deformed into the corresponding boundary components). But one can easily represent elements a\d\,.. .,a\db by embedded nonseparating loops. It follows that 7ri(5) can be generated by elements represented by nonseparating embedded loops. This remark will be used in our proof of the Dehn-Lickorish theorem (see the proof of Theorem 4.2.C). If C is a circle on S or, more generally, union of several disjoint circles, we can cut S along C and get a new surface, which will be denoted by ^c • If C is a nonseparating circle, then Sc is connected. If C is a separating circle, then Sc consists of two components. Note that we can always reconstruct S from Sc by gluing along the boundary components of Sc resulting from C. More generally one can cut S also along proper arcs and, more generally, disjoint unions of circles and proper arcs. If (proper) arcs are involved, the result of cutting is a smooth surface with comers. Suppose now that C is a nonseparating circle. Then Sc is a connected surface having two more boundary components than S. The additivity of the Euler characteristic implies that Sc has the same Euler characteristic as S. Hence, by the classification of surfaces, Sc is determined up to a diffeomorphism by S (and the fact that C is nonseparating). LEMMA 2.3.A. If C, C' are two nonseparating circles on S, then there is a diffeomorphism f: S ^ S such that f(C) = C'.

PROOF. By the remarks preceding the lemma, Sc and Sc are diffeomorphic. Let us fix an orientation of S\ it induces an orientation of both Sc and Sc- Since every orientable surface obviously admits an orientation-reversing diffeomorphism, we can always choose an orientation-preserving diffeomorphism F \ Sc ^^ Sc. Let C_ and C+ be two boundary components of Sc resulting from C, and let C_ = F(C-), C^ = ^(C+). Let I:C-^ C^ and I' '.C'_^^ C^ be two gluing diffeomorphisms giving S back from Sc and Sc', respectively. If F agrees with these gluings, i.e., if F o / = / ' o F on C-, then F gives rise (by gluing) to a diffeomorphism f: S -^ S such that / ( C ) = C . Note that both / and / ' are orientation-reversing (assuming that C-, C+, CL, C;^ are oriented as boundary components of Sc, Sc) and F is orientation-preserving on both C_ and C+. It follows that diffeomorphisms F o /, / ' o F : C- -^ C;^ are both orientation-reversing. Since every two orientation-reversing (as also every two orientation-preserving) diffeomorphisms of a circle are isotopic, we can find a diffeomorphism F ' : C+ ^- C;^ isotopic to F | C-f and such that F ' o / = / o F I C_. By extending the isotopy between F | C+ and F ' to an isotopy of F (and without changing F | C_), we get a diffeomorphism F: Sc ^^ Sc which agrees with gluings. As we noted above, this is sufficient to complete the proof. D

530

N.VIvanov

If C is nonseparating, then there is a circle C transversely intersecting C at exactly one point. (To construct such a C^ take a small arc / intersecting C transversely at one point in the interior of / and connect the two ends of / by an arc in *S \ C.) In particular, the homological intersection number C • CMs ± 1 . This immediately implies that C is nontrivial. LEMMA 2.3.B. A circle C contractible in S bounds a disc in S. A circle C homotopic to a boundary component B of S, bounds, together with B, an annulus in S. In particular, a circle is non-trivial if and only if it does not bound a disc in S and does not bound an annulus in S together with a boundary component.

First, notice that a trivial circle C is always separating by the remarks preceding the lemma. For such a C, we can represent S as the union of two surfaces S\, ^2 intersecting only along their common boundary component C. If C is contractible in 5, then van Kampen's theorem (together with the basic properties of the amalgamated products) implies that C is contractible in one of the surfaces S\, S2, say in ^i. In view of 2.2, any embedded loop with image C is freely homotopic to a loop representing di or df for some generator di of n\(S\). Because C is contractible in S{, this implies that di = 1, Obviously, this is possible only if the genus of S\ is 0 and ^i has only one boundary component (namely, C), i.e., only if ^i is a disc. In this case C bounds a disc in S, namely S\. A similar argument applies if C is homotopic to a boundary component. We leave it to the reader. D PROOF.

2.4. Pants decompositions With few exceptions, a surface has negative Euler characteristic (by the classification of surfaces). The simplest such surface is a disc with two holes, often called a pair ofpants, or simply pants; see Figure 1(a). As the next theorem shows, pairs of pants are the elementary building blocks of all surfaces of negative Euler characteristic. THEOREM 2.4. A. If S is a surface of negative Euler characteristic {i.e., if S is not a sphere, a torus, a disc, or an annulus), then there is a collection of disjoint circles on S such that the result Sc of cutting S along their union C is a disjoint union ofpairs of pants. In other words, C decomposes S into pairs of pants.

Using the classification of surfaces, one can easily exhibit the required collection of circles in each case. Let us outline a proof independent of the classification. Choose a PROOF.

(a)

(b) Fig. 1.

Mapping class groups

531

Morse function f: S ^ K on S such that / is equal to 0 on 95, / is nonnegative and the values of / at different critical points are different. Let 0 = ao < a\,... < Un be a sequence of noncritical values of / such that every interval [a/_i, a/], 1 < / < «, contains exactly one critical value of / and Im / c [0, «„). Consider a component of the preimage /~^ ([fl/_i, a/]). If it does not contain a critical point of / , it is an annulus. If it contains a critical point of / of index 0 or 2, then it is a disc. Finally, if it contains a critical point of / of index 1, then it is easily seen to be a disc with two holes. So, if we cut S along the union of the components of all preimages f~^(ai)J ^ I, the resulting surface will consist of discs, annuli, and pairs of pants. We can simplify this decomposition of S into discs, annuli, and pairs of pants in the following manner. Note that gluing a disc to a boundary component of a disc, annulus or a pair of pants results in a sphere, a disc, or an annulus respectively. Also, gluing an annulus to a boundary component of any surface does not change it up to a diffeomorphism. Hence, by repeatedly gluing discs and annuli to other pieces of the decomposition, we will eventually reach a decomposition of S either into several pairs of pants, or into two discs glued together (in this case 5 is a sphere), or into an annulus glued to itself (in this case 5 is a torus), or into just one disc or annulus. To complete the proof, note that a gluing operation corresponds to removing a circle from our collection of cutting circles. D This theorem can be used to give an alternative proof of the classification of surfaces. Let us cut S along the union of several circles into a finite collection of pairs of pants, as in the theorem. Then let us reassemble S from this collection in the following manner. Start with some pair of pants and glue to it the other pairs of pants, one at time, and only along one boundary component at a time, as long as this is possible. At every stage we will have, as one can easily see by induction, a disc with several holes. When no further such gluing is possible, the remaining gluings identify some pairs of boundary components of our disc with holes. If there are g such pairs and b boundary components not in any such pair, then the result of the gluing of these pairs is, obviously, a sphere with g handles and b holes, i.e., a surface of genus g with b boundary components. Sometimes it is useful to decompose pairs of pants further. Namely, three nontrivial arcs, shown in Figure 1(b), decompose a pair of pants into two "topological hexagons" (which are smooth manifolds with comers and are homeomorphic, but not diffeomorphic to a disc). Combining this decomposition with Theorem 2.4.A, we can decompose S into such topological hexagons. From the perspective of 3-dimensional topology, nontrivial circles and arcs are analogues of incompressible surfaces, and the above decomposition of S into hexagons is an analogue of the Haken decomposition of a Haken manifold into topological balls. In fact, since pairs of pants are simple enough, one rarely needs to use nontrivial arcs to decompose a surface into simpler pieces. Note that from the point of view of this analogy all surfaces, with the exception of discs and spheres, are Haken. 2.5. Geometric structures on surfaces By a geometric structure on a surface S we understand simply a Riemannian metric on S having constant curvature and geodesic boundary. By scaling the metric, if necessary, we

532

N.VIvanov

can assume that its curvature is equal to 1, 0 or —1. It turns out that every surface admits a geometric structure. For surfaces of nonnegative Euler characteristic this is very simple. For a sphere, it is sufficient to take the metric of the standard unit sphere in R^. For a disc, it is sufficient to take the metric of a hemisphere of a standard sphere. For an annulus, one may take a product metric on 5' x [0,1]. For a torus, one may take the metric on R^/Z^ induced from the standard Euclidean metric on R^. In particular, spheres and discs admit metrics of (constant) positive curvature, and annuli and tori admit metrics of curvature 0. The really interesting case is the case of surfaces of negative Euler characteristic. They all admit hyperbolic structures, i.e., geometric structures with metric of curvature —1; see Theorem 2.5.B below. Geometric structures on surfaces can be used on two levels. On the first level, the very existence of a geometric structure is useful. For example, fixing a geometric structure on a surface S allows us to represent isotopy classes of circles by geodesies; for hyperbolic structures such representatives are unique. Moreover, the following lemma holds. LEMMA 2.5.A. Suppose that S is endowed with a geometric structure. Then every nontrivial circle on S is isotopic to a geodesic circle on S. If S is endowed with a hyperbolic structure, then such a geodesic circle is unique. If two nontrivial circles are disjoint, then any two geodesic circles isotopic to them are either disjoint or equal.

We refer to [53, Expose 3, §§I and III] for a proof. On a second level, one considers all possible geometric structures on S simultaneously. It is convenient to identify geometric structures on S which differ one from another in a trivial way; namely, to identify two geometric structures if one is the pull-back of the other by a diffeomorphism S ^^ S isotopic to the identity. The resulting set of equivalence classes is the Teichmiiller space of S. We explore this point of view in Section 5. THEOREM 2.5.B. If a surface has negative Euler characteristic, then it admits a hyperbolic structure. PROOF. There are (at least) two ways to prove this result. The first one works best for closed surfaces S. Represent 5 as a result of the standard identification of sides of a 4^gon; cf. 2.2. One can realize this Ag-gon as a regular polygon on the hyperbolic plane with all angles equal to 27t/4g = njlg. Note that the hyperbolic plane has a canonical Riemannian metric of constant negative curvature (in the upper half plane model, it is the metric (dx^ -h dy^)/y^ on {(jc, y) 6 R^: y > 0}); this metric induces a Riemannian metric on S because the lengths of glued edges are equal and the sum of angles is equal to In (notice that all vertices of our 4g-gon are identified in the gluing process). Clearly, this metric on S has constant negative curvature. Note that it is not necessary to consider a regular 4g-gon; a weaker condition would suffice. For the details of this construction we refer to [217, Sections 5.5 and 5.6]. Now, one can use Lemma 2.5.A to deal with the surfaces with boundary. Given a surface with boundary 5, we can endow the double d^ of S with a hyperbolic structure. Recall that d5 is the result of gluing S with a copy of itself along the boundary; in particular, d^ D 5. Lemma 2.5.A implies that 95 is isotopic in d5 to a union of disjoint geodesic circles. If we

Mapping class groups

533

cut dS along this union, we get two surfaces diffeomorphic to S, each of them endowed with a hyperboUc structure. Another construction works equally well both for closed surfaces and for surfaces with boundary and is based on Theorem 2.4.A. Let us prove first that a pair of pants P admits a hyperbolic structure such that all three boundary components have the same length. To this end, we represent P as the union of two topological hexagons as in Figure 1(b). Let us realize these hexagons as two (isometric) regular hexagons in the hyperbolic plane with all angles equal to n/2. Then gluing P from these hexagons endows P with a hyperbolic structure (one should take isometrics as the gluing maps; this is possible because lengths of edges are all equal); the boundary is geodesic because angles of hexagons are equal to 7t/2. Clearly, the lengths of all boundary components of P are equal to twice the length of an edge of our hexagons; in particular, the lengths of all boundary components of P are equal. Now, given a surface S of negative Euler characteristic, we can decompose S into pairs of pants as in Theorem 2.4.A. Let us endow all pairs of pants of this decomposition with the hyperbolic structure we just constructed. Since the lengths of all boundary components of these pairs of pants are equal, one can take isometrics as gluing maps and then these hyperbolic structures on pieces of S induce a hyperbolic structure on S itself. For the details, and a more general version of this approach, we refer to [29, Section 1.7 and Chapter 3]. It can be easily generalized to give a description of all hyperbolic structures on S; see [29, Chapters 3 and 6], and [53, Expose 7 and Section 5]. D 2.6. Spaces of dijfeomorphisms and embeddings: generalities For an orientable manifold M we denote by DifP(M) the group of all diffeomorphisms M ^^ M, and by Diff(M) the subgroup of all orientation-preserving diffeomorphisms. (Since we are interested mainly in the groups Diff(M), we chose for them a simpler notation.) For X C M we denote by DifP(MfixZ) the subgroup of diffeomorphisms f :M ^^ M fixed on X, i.e., such that f(x) = x for all x e X. We denote by Diff(MfixX) the intersection Diff (MfixX) Pi Diff(M). The most important case is that of X = dM; we denote the groups Diff'(M fix dM) and Diff(M fix dM) by Diff>(M fix 3) and Diff(M fix 3) respectively. If A^ is a submanifold of M, we denote by Emb"^(A^, M) the set of all embeddings N -> M which can be extended to a diffeomorphism M ^^ M. If N has codimension 0 in M (the main case is that of a disc in the interior of M), then it makes sense to speak about orientation-preserving embeddings N ^^ M, and we denote by Emb(A^, M) the subset of all orientation-preserving embeddings. By Sub(A^, M) we denote the set of submanifolds of M diffeomorphic to N by SL diffeomorphism which extends to M (i.e., the set of all images of embeddings from Emb^(A^, M)). As usual, all diffeomorphisms and embeddings are assumed to be of the class C ^ , and the above groups of diffeomorphisms and sets of embeddings are considered together with their C^-topology. Sub(M, A^) is considered with the topology of a quotient space of Emb^(A^, M). As is well known, groups of diffeomorphisms with the C^-topology are topological groups. Also, for a submanifold N of M the natural map Diff^(M) -^ Emb'^(A^, M) given by the restriction of diffeomorphisms to A^ is continuous.

534

N.V.Ivanov

It is easy to see that if M is a surface, then Mod^ = 7ro(Diff(5)), Mod^ = 7ro(Difr(5)) a n d X 5 = 7ro(Diff(5fixa)). 2.6.A. Let N be submanifold of M such that dN intersects dM along several components of dN and N is transverse to dM. Then the natural map Diff^(M) -> Emb^(A^, M) is a Serre fibration. If N is of codimension 0 in M, then the natural map Diff(M) -> Emb(A^, M) is a Serre fibration. If N is of codimension 0 in M and is contained in the interior of M, then the natural map Diff(M fix9) -^ Emb(A^, M) is also a Serre fibration. THEOREM

PROOF. Recall that a continuous map is a Serre fibration if the homotopy lifting property holds for cubes. For Diff^(M) -^ Emb'^CA^, M) the homotopy lifting property for 0-dimensional cubes amounts to the extension of isotopies of A^ in M to isotopies of the whole M. This extension property is well known and easy to prove; see for example, [128, Theorem II.5.2]. The homotopy lifting property for higher-dimensional cubes is essentially a multiparameter version of this isotopy extension property and can be proved by a direct generalization of the standard proof of the usual isotopy extension property. Also, the second statement of the theorem obviously follows from the first. The last statement of the theorem can be proved in exactly the same manner as the first one. D

2.6.B. The natural map Em\f{N,M) -> Sub(A^, M), Emb(A^,M) -^ Sub(A^, M), given by the formula f \-^ f{N), are Serre fibrations. THEOREM

This theorem can be proved using a variant of the ideas of the proof of Theorem 2.6. A. The fibers of the fibrations Diff^(M) -^ Emb'^CA^, M) and Diff(M) -^ Emb(A^, M) are easy to identify. They are Diff^(MfixA^) and Diff(MfixA^) respectively. In addition, if A^ is a submanifold of codimension 0 contained in the interior of M, then we can identify DifTCM fixA^) and Diff(MfixA^) with DifPCM \ iniNfixdN) and Diff(M \ iniNfixdN) respectively. The fibers of the maps Emb^(A^, M) -^ Sub(A^, M), Emb(A^, M) -^ Sub(A^, M) are, obviously, Diff^(A^) and Diff(A^) respectively. Now, let us fix a point m e M and consider the space FM of all orientation-preserving isomorphisms TmM ^^ TxM,x e M (where TxM is, as usual, the tangent space to M at x). There is an obvious projection FM -> M; moreover, FM is a principal GL^(«, R)-bundle over M, where n = dimM. Let d: Diff(M) -^ FM be the map assigning to a diffeomorphism M ^- M its differential at m. If A^ is a submanifold of codimension 0 and m e N, then there is a similar map d: Emb(A^, M) -^ FM. THEOREM 2.6.C. The maps d: Diff(M) -^ FM and d: Emb(A^, M) -^ FM are Serre fibrations. If N is a disc of codimension 0 in the interior of M, then the second map is a weak homotopy equivalence.

We may consider points of FM as framed points in M. Then the homotopy fitting property for d becomes a version of the (multiparameter) isotopy extension property for isotopies of points, which is not much more difficult to prove than the usual isotopy extension property. The second statement of the theorem is essentially a multiparameter version of a well known theorem to the effect that there is exactly one isotopy class PROOF.

Mapping class groups

535

of orientation-preserving embeddings of a codimension 0 disc in an orientable manifold. See, for example, [128, Corollary III.3.6]. Again, it is easy to add parameters to the usual proof. n If dimM = 2, it convenient to fix a Riemannian metric on M and consider the unit tangent bundle UT(M) instead of FM. Let us fix a vector v eT^M. Then the evaluation at V followed by the normalization defines a map FM -^ UT(M), which is a homotopy equivalence (if dimM = 2). Also, define u : Diff(M) -^ UT(M) as the map assigning to a diffeomorphism M ^- M the normalized image of v under its differential at m. Similarly define u : Emb(A^, M) -^ UT(M). THEOREM 2.6.D. The maps u : Diff(M) -^ UT(M) and u : Emb(A^, M) -> UT{M) are Servefibrations.If N is a disc of codimension 0 in the interior ofM {and dim M = 2), then the second map is a weak homotopy equivalence. PROOF. The first statement is similar to the first statement of Theorem 2.6.C and, in fact, holds in all dimensions. The second statement follows from the second statement of Theorem 2.6.C and the fact that the above map FM -^ UT(M) is a homotopy equivalence. D

The results of this section go back to Cerf [31 ] and, for closed manifolds, Palais [ 196]. In fact, they proved that our Serre fibrations are actually fiber bundles. But we need only the Serre fibration property, because we are interested only in the homotopy groups of spaces of diffeomorphisms and embeddings (in fact, only in TTQ and TTI ), and this property is much easier to prove. 2.7. Spaces of diffeomorphisms and embeddings: surfaces In this section, we describe the weak homotopy type of some spaces of diffeomorphisms of surfaces and embeddings of circles. While all these results are known to be true for the usual homotopy type instead of the weak one, the results about the weak homotopy type are easier to prove (in particular, there is no need to know that Serre fibrations from 2.6 are fiber bundles), and only they are needed for the applications we have in mind. We start with two preliminary results about 1-dimensional manifolds. LEMMA 2.7.A. Diff(/)^ fix 9) is contractible, where D^ is the \-dimensionaldisc. We may assume that Z)^ = [0,1]. Now, if / : [0,1] ^- [0, 1] is a diffeomorphism fixed on {0,1}, then x i-> (1 — t)f{x) + ^x is a diffeomorphism for every ^ e [0, 1], because its derivative is always > 0. D PROOF.

COROLLARY 2.7.B. Diff(5'^) is homotopy equivalent to S^ and, moreover, contains S^ as a deformation retract. PROOF. Note that S^ is naturally contained in Diff(5'') as the group of rotations. Fix X eS^ and consider the Serre fibration Diff(5^) -^ Emb({x}, S^). Its base can be obviously identified with 5'^ and S^ C Diff(5^) defines a section. Using this section, it is easy

536

N.V Ivanov

to see that this Serre fibration is actually a fiber bundle. If we notice that its fiber is homotopy equivalent to Diff(D^ fix 9), the corollary follows. D The first basic result about the groups of diffeomorphisms of surfaces is the following theorem of Smale [215]. THEOREM

2.7.C. Diff(D^ fix9) is contractible, where D^ is the 2-dimensionaldisc.

In that follows, we understand by a weak deformation retract of a topological space X a subspace Y C X such that the inclusion F -^ X is a weak homotopy equivalence. The following corollary is also due to Smale [215] (who actually proved it with all adjectives "weak" omitted). COROLLARY 2.7.D. Diff(D^) is weakly homotopy equivalent to a circle. DiffC^^) is weakly homotopy equivalent to SO(3) and, moreover, contains SO(3) as a weak deformation retract.

Consider the (Serre)fibrationDiff(D^) -^ Diff(9D^). Its base is homotopy equivalent to S^ by Corollary 2.7.B and its fiber is homotopy equivalent to a point by Theorem 2.7.C. The first statement follows. Now, fix a disc D^ c S^ and consider the fibration Diff(5^) -^ Emb(D2, S^). By Theorem 2.6.D its base Emb(D^, S^) is weakly homotopy equivalent to the unit tangent bundle of 5^, which can be identified with S0(3). The fiber is Diff(5^ fix D^), which can be identified with Diff(5^ \ int D^ fixdD^). But S^ \ int D^ is a disc, hence by Theorem 2.7.C the fiber is contractible. It follows that Diff(5^) is weakly homotopy equivalent to S0(3). Moreover, the fibration u:Diff(S^) -^ UT(S^) = S0(3) from 2.6 is a weak homotopy equivalence. The inclusion of SO(3) in Diff(5'^) as the subgroup of rotations defines a section of this fibration. Clearly, the image of this section, i.e., SO(3), is a weak deformation retract of Diff(5'^). D PROOF.

2.7.E. Let A be an annulus (i.e., a manifold diffeomorphic to S^ x [0, 1]). Then Diff(Afix9) has weakly contractible components, and the group of components 7ro(Diff(A fix 9)) is isomorphic to Z. COROLLARY

Consider the fibration Diff(D^ fix 9) -> Emb(5^, D^), where D^ is a disc and B^ is a disc in the interior of D^. Its total space is contractible by Theorem 2.7.C. By Theorem 2.6.D its base is weakly homotopy equivalent to the unit tangent bundle of D^ and, hence, to S\ Its fiber is Diff{D^ \ int B^ fix 9), which can be identified with Diff(A fix 9). The corollary easily follows from these facts and the homotopy sequence of our fibration. D PROOF.

2.7.F. Let T = S^ x S^ be a torus. The components o/Diff(r) are weakly homotopy equivalent to T and, in fact, contain T as a weak deformation retract.

THEOREM

Note that since T is a Lie group, T is naturally contained in Diff(r). In fact, this subgroup T c Diff(r) is a (weak) deformation retract of the component of the identity of Diff(r).

Mapping class groups

537

THEOREM 2.1 .G. If S is a surface of negative Euler characteristic, then the components of Diff(5 fix 9) and of Diff(5) are weakly contractible.

Theorems 2.7.F and 2.7.G (in fact, with adjectives "weak" omitted) are due to Earle and Eells [46]. The proofs in [46] are analytical and are based on the theory of Beltrami equations. The corresponding results about spaces of homeomorphisms are due to Hamstrom [73,74]; her proofs are purely topological. The corresponding results in the piecewise linear category were obtained by Scott [210]. Nowadays the easiest way to obtain Theorems 2.7.F and 2.7.G is, probably, to consider them as 2-dimensional versions of 3-dimensional results of Hatcher [88] and the author [94] about Haken manifolds (cf. end of 2.5). Unfortunately, such an approach is not written down yet. THEOREM 2.7.H. Let S be a surface of negative Euler characteristic and let C be a nontrivial circle on S. Then the components o/Emb(C, S) are weakly homotopy equivalent to a circle and the components 6>/Sub(C, S) are weakly homotopy equivalent to a point.

This theorem can be deduced from the above results about spaces of diffeomorphisms using fibrations from 2.6. Alternatively, if one proves Theorems 2.7.F and 2.7.G along the lines of theorems of [88] and [94] about Haken manifolds, then this theorem is, essentially, the main step of the proof. Finally, we give an application of these theorems which will be needed later (see 6.3). THEOREM 2.7.1. Let R be a connected subsurface of a connected surface S of negative Euler characteristic. Suppose that the boundary of R is connected and nontrivial in S. Then the natural map MR -^ Ms, given by the extension of diffeomorphisms of R fixed on dR by the identity to diffeomorphisms ofS, is injective. PROOF. Let T be the closure of S \ R. Clearly, T is connected. The natural map Diff(Sfixa) -» Emb(r, 5) is a Serre fibration with fiber Biff(Rfixd). The map 7io(Diff(R fix d)) -^ 7ro(Diff(5'fix9)) induced by the inclusion of the fiber in the total space is nothing other as our map MR -^ Ms- In view of the homotopy sequence of this fibration, this map will be injective if TTI (Emb(r, S)) = 0. Note that since dR is nontrivial, T has negative Euler characteristic and hence 7t\ (Diff(r)) = 0. Let us consider now the natural map Emb(r, S) -^ Sub(r, S) from Theorem 2.6.B, which is a Serre fibration with fiber Diff (7). In view of the homotopy sequence of this fibration, TC\ (Emb(r, S)) =0 if TTi (Sub(r, 5)) = 0. But a submanifold of S diffeomorphic to T is determined by its boundary (may be up to two possibilities, if R is diffeomorphic to T). It follows that 7ri(Sub(r, 5)) = 7ri(Sub(ar, S)). The latter group is 0 in view of Theorem 2.7.H. This completes the proof. D

2.8. Gluing discs to a surface If a surface has nonempty boundary, then we can construct a closed surface or a surface with a smaller number of boundary components by gluing discs to boundary components.

538

N.VIvanov

Very often this gluing procedure allows us to reduce theorems about surfaces with possibly nonempty boundary to the case of closed surfaces, or use the induction on the number of boundary components. In this section we discuss some basic constructions and results used in such arguments. Let /? be a surface with nonempty boundary and let Q be the result of gluing discs D\,..., Dnion different boundary components of R. Given a diffeomorphism f: R-^ R preserving setwise all components of dR, there exists an extension f '.Q^- Qof f. Such an extension is obviously not unique, but it is unique up to isotopy, because the spaces Diff(D/ fix 9) are connected. Hence, extension of diffeomorphisms from Rio Q defines a map p : PMod/? -^ PModg. Clearly, p is a homomorphism. We are interested in particular in its kernel, and to understand it we need the notion of the pure braid groups of surfaces and the properties of these groups discussed below. For any surface 5, consider S^^^ = {iy\,..., ym) ^ S^: yi / yj for / y^ j}. The pure braid group PBm (S) is defined as 7T\ (S^^^), where we tacitly assume that some base point (jci,..., x^) G S^"^^ have been chosen. Let S(i) = S \ {x\,.. .,Xi-\,X/+1,.. .,Xni} (note that S(i) is a noncompact surface). Let Xi be the base point of S^y The map 5(/) -^ S^^^ given by the formula x i-^ (x\,.. .,jc/_i,x,x/+i,. ..,Xm) induces a map 7t\(S(i),Xi) -^ 7t\(S^'^^) = PBfn(S). THEOREM

2.8.A.

The group PBmiS) is generated by the images of the above maps

Use induction by m and the natural fibrations S^^^ -^ S^^~^^ given by the formula ( y i , . . . y / ) ^ (ji,...,y/-i)•

PROOF.

We are mainly interested in the group PBn{Q). We will assume that the base point (x\,.. .,Xn) G Q^^^ has been chosen in such a way that Xj G intD/, 1 ^ / ^n. Let Qi be the result of gluing only the disc D/ to R. Clearly, Qi is contained in 2(/) and hence natural maps n\(Qi,Xi) -^ PBn{Q) are defined. In fact, Qi is a deformation retract of 2(/). COROLLARY 2.8.B. The group PBn(Q) is generated by the images of the natural maps ni(Qi,Xi)^PB,(Q).

Now we construct a homomorphism j :PBn{Q) -^ PMod/?. Let ^ G PBn{Q)\ represent ^ by a loop in Q^^^K Such a loop may be considered as an isotopy of the 0-dimensional submanifold{jci, ...,Xn] of 2 Let us extend this isotopy to an isotopy [f'-Q^Glo^/^i of the whole manifold Q\ in particular, /o = xdQ, f\(xi) = x/ for all /. We can easily arrange that, in addition, f\ (D/) = Z), for all /. (It follows from Theorem 2.6.C that the space of discs in a surface, containing a given point in the interior, is connected.) Then f\ preserves R and preserves all boundary components of R. We take as j(P) the isotopy class of the restriction of f\ on R. One can easily check that j is correctly defined (using Theorem 2.6.C again). We leave it as an exercise to check that j is a homomorphism. If n = 1, then PBn(Q) = PB\(Q) is nothing more than the fundamental group 7T\(Q) and j is a homomorphism 7t\(Q) -^ PMod/?. The first main property of j is contained in the following theorem.

Mapping class groups

539

2.8.C. The homomorphism p is surjective and Kerp = 11117; ^^ other words, the sequence THEOREM

PBn(Q) —> PMod/? —> PModQ —> 1 is exact. If Q has negative Euler characteristic, then j is injective; in other words, the sequence 1 —> PBn(Q) —> PMod/? —> FModQ —> 1 is exact. PROOF. Let us prove the surjectivity of p : PMod/? -^ PModg first. Represent an element a of PMod^ by a diffeomorphism g.Q^^ Q. Connectedness of the spaces of (orientationpreserving) embeddings of discs into a surface easily implies that g is isotopic to a diffeomorphism g^ such that g\Di) = Di,l ^i ^n. Let y be the isotopy class of the restriction of g^ to R. Clearly, p(y) = a. This proves the surjectivity of p. Now, let y e Ker p. Let us represent y by a diffeomorphism f: R ^^ R and extend / to a diffeomorphism f :Q^^ Q- We may assume that f\xi) = x/, 1 ^ / ^ n. Since y e Kerp, the diffeomorphism /^ is isotopic to the identity idg. Let {//: Q —^ Q}o^t^\ be some isotopy between /^ = /^ and / ( = id^. Then t i-> ( / / ( x i ) , . . . , fl(xn)) defines a loop in Q^^^ with the base point (jci,...,x„) and, hence, an element p e 7t\(Q[n\) = PBn(Q). Obviously, j(P) = y. This proves that Ker p Clmj. The opposite inclusion follow directly from the definitions. Suppose now that the Euler characteristic of Q is negative. Let p e Ker 7. Let us represent y^ by a loop in Q^"^ and construct an isotopy {ft'- Q -^ Glo^/^i as in the definition of j . In particular, f\ (Di) = (Di) for all /. Since fi e Ker7, the restriction of f\ on R is isotopic to id/?. It follows that after changing the isotopy, if necessary, we may assume that the restriction of f\ on R is equal to id/?. Next, in view of Theorem 2.7.C, we may assume that /i = idQ. Now, {ft:Q^ Qh^t^i defines a loop in Diff(|2). By Theorem 2.7.G this loop is contractible. Clearly, this implies that the original loop representing fi is contractible, and hence p = I. •

This theorem is due to Birman [14]. See also [16, Theorem 4.2]. In general, Ker 7 is isomorphic to the quotient of the group PBn(Q) by its center, which can be easily identified; see [16, Chapter 4]. Now, let a e Mod/?. Suppose that some (and then every) diffeomorphism f : R -^ R representing a preserves (setwise) the union of circles dDi, I ^ i ^ n. Then / can be extended to a diffeomorphism f '.Q ^^ Q. Such extension is unique up to isotopy (cf. the construction of p). We may assume that f\xi) = XaH) for some permutation a :{\,.. .,n} ^^ {!,...,«}. Extension /^ is unique up to isotopy even in the class of such diffeomorphisms (this follows from the fact that discs are connected and the results of 2.6). Such an extension / ' induces a map Q^"^ -^ Q^^^ mapping the base point * = {xi,..., x„} to a(*) = {xa(\),..., X(j(^)}. In particular, f^ induces a homomorphism of the fundamental groups 7r\{Q[n],^) -^ 7T\(Q[n],cr(^)). It is easy to see that this homomorphism depends only on Of. By a slight abuse of notations, we will denote it, even if a ^ id, by

540

N.VIvanov

Of* :PBn(Q) -^ PBniQ) (keeping in mind that the second pure braid group is defined using a different base point). Note that, obviously, a* maps the image of 7X\{Qi,Xi) in PBniQ) to the image of The homomorphisms a* and j are related by the following fundamental formula:

which follows immediately from the constructions of a* and j . As the last application of the extension of diffeomorphisms from Rio Q, consider the situation when only one disc, D\, is added to R. Then the extension of diffeomorphisms defines a map Mod'^ -^ 7ro(Diff(g fixxi)), where Mod'^ is the subgroup of Mod/? consisting of isotopy classes of diffeomorphisms preserving 9Di. It turns out that this map is an isomorphism. (Again, this follows from 2.6.)

2.9. The Dehn-Nielsen-Baer theorems For a group TC, let Aut(7r) be the group of automorphisms of it, and let Inn(7r) be the subgroup of inner automorphisms of TT. A trivial check shows that Inn(TT) is a normal subgroup of Aut(7r). The quotient group Aut(7r)/Inn(TT) is denoted by Out(7r) and called the outer automorphisms group of n. These groups naturally show up when one deals with homotopy self-equivalences of a (connected) space X with n\ {X) = n. Namely, any selfmap f :X ^^ X defines a homomorphism /* : TTI (X) -^ n\ (X) and if / is a homotopy equivalence, then /* is an isomorphism. But, if we do not assume a base point to be fixed, /* is well defined only up to an inner automorphism of 7t\ (X) = n. Hence, a homotopy equivalence f :X ^^ X defines an element /* e Out(7r). Obviously, /* depends only on the homotopy class of / . Now, let X = 5 be a compact (connected) orientable surface. In view of the above remarks, there is a natural homomorphism Mod^ -^ Out(7ri (S)). If S has nonempty boundary, then the elements of the image of this homomorphism obviously preserve the peripheral structure of 7t\(S) from 2.2. THEOREM 2.9.A. If S is closed and is not a sphere, then the natural homomorphism Mod^ -^ Out(7ri {S)) is an isomorphism. If S has nonempty boundary and negative Euler characteristic (i.e., S is not a disc or an annulus), then the natural homomorphism Mod^ -^ Out(7ri(5')) is an isomorphism onto the subgroup of 0\xi{7t\{S)) consisting of elements preserving the peripheral structure.

One can complement this theorem by a description of the subgroup of Out(7ri {S)) corresponding to Mods. Suppose first that S is closed (and is not a sphere). (For a review of the cohomology of groups, used below, see 6.1.) Then S is an Eilenberg-MacLane space, and hence H^{7T\{S),Z) = H^(S,Z) = Z. The group Out(7ri(5)) naturally acts on H^ini (S), Z), because inner automorphisms act trivially on cohomology groups. Let Out"^(7ri(5)) be the subgroup of Out(7ri(5')) consisting of elements acting trivially on H^(n\ (S), Z). Clearly, the image of Mod^ under the homomorphismMod^ -^ Out(7ri (S))

Mapping class groups

541

is contained in Out^(7ri (S)). If S has nonempty boundary (and negative Euler characteristic), then the image of Mod^ in Out(7ri (S)) is obviously contained in the subgroup of Out(7ri (5)) consisting of elements preserving the oriented peripheral structure from 2.2. THEOREM 2.9.B. If S is closed and is not a sphere, then the natural homomorphism Mod^ -^ Out^(7ri {S)) is an isomorphism. IfS has nonempty boundary and negative Euler characteristic, then the natural homomorphism Mods -^ Out(7ri(5')) is an isomorphism onto the subgroup ofO\xi{7T\ (S)) consisting of elements preserving the oriented peripheral structure.

For closed S, the surjectivity part of these theorems is due to Dehn, who did not pubUsh his proof, and to Nielsen [191], who published a proof partially based on Dehn's ideas. Another proof was suggested by Seifert [211]. The injectivity part is due to Baer [3]. The surjectivity part for surfaces with boundary was proved by Magnus [148], and the injectivity part - much later - by Zieschang [244]. Proofs and a detailed discussion of these results can be found in [245]; see [245], Theorem 3.3.11 (surjectivity for closed surfaces following Seifert), Theorem 5.7.1 (surjectivity for surfaces with boundary) and Theorem 5.13.1 (injectivity). In principle, these results give a purely algebraic description of mapping class groups and allow one to reduce every question about them to a purely algebraic question. Surprisingly, this reduction turned out to be not very successful. Probably, the only algebraic property of the mapping class groups which for a long time could be proved only in terms of the outer automorphisms groups is the residually finiteness. 2.9. C . The groups Mod^ are residually finite, i. e.,for every nontrivial element f G Mod^, / 7^ 1, there is a homomorphism h : Mod^ -> G onto a finite group G such that

THEOREM

h(f)^L This theorem is due to Grossman [68], whose proof was based on the description of mapping class groups in terms of outer automorphisms groups and fairly complicated combinatorial group theory arguments. A more conceptual proof, still based on the description in terms of outer automorphisms groups was given by Bass and Lubotzky [6]. Now a simple direct proof, not based on the results of Dehn-Nielsen-Baer, is available; see [107] or [108, Exercise 1]. In one case, namely, if 5' is a closed torus, the Dehn-Nielsen-Baer theorems provide a complete and efficient description of Mod^ and Mod^. In this case 7t\(S) is AbeUan and, moreover, is isomorphic to Z^, and hence Oui(n\(S)) = Aui(7T\(S)) = Aut(Z^) = GL2(Z). This chain of isomorphisms clearly maps the subgroup of Out+(7ri(5)) to SL2(Z). It follows that Mod^ = GL2(Z) and Mod^ = SL2(Z) for a closed torus 5. 3. Complexes of curves Complexes of curves are, probably, the most fundamental geometric objects on which the mapping class groups act (but the Teichmiiller spaces are, certainly, the most important). They were discovered by Harvey [86,87], and have played an ever increasing role since

542

N.VIvanov

then. In this section we give the definition and discuss the basic properties of them. We prove that complexes of curves are connected, and do this in a way which can be generaUzed to prove much stronger homotopy properties of them. After this, we discuss the homotopy type and the hyperboUc properties of them. Later sections will amply illustrate the usefulness of complexes of curves. 3.1. Definitions Complexes of curves are simphcial complexes in the sense of [216, Chapter 3] or [28], for example. Thus, a simplicial complex consists of a set of vertices and a set of simplices. Simplices are nonempty finite sets of vertices, subject only to the following two conditions: a nonempty subset of a simplex is a simplex; every vertex belongs to some simplex. The dimension of a simplex is the number of vertices in it minus 1. One-dimensional simplices play a special role and are called edges. The vertices of the complex of curves C(S) of a compact orientable surface S are the isotopy classes of simple closed curves (as usual, we call them circles) on 5, which are non-trivial in the sense of 2.3, i.e., not contractible in S into a point or into the boundary dS. We denote the isotopy class of a circle C by (C). A set of vertices {yo, • • •, y«} is declared to be a simplex if and only if yo = {Co),..., yn = (Cn) for some pairwise disjoint circles C o , . . . , C«. The extended mapping class group Mod^ acts on C(5) in an obvious way: the isotopy class / e Mods of a diffeomorphism F :S ^^ S maps (C) to {F(C)). C(S) can be defined in another way. According to 2.5, we can equip S with a Riemannian metric of constant curvature with geodesic boundary. Then we can take as the set of vertices of C(S) the set of geodesic circles in S\dS and as simplices the sets of pairwise disjoint geodesic circles. In view of Lemma 2.5.A, this definition is equivalent to the previous one. This definition makes clear that a set of vertices is a simplex if and only if all its 2-element subsets are simplices (i.e., any two its vertices are connected by an edge). In the language of the theory of buildings (cf. [28], for example) this means that C(S) is ^flag complex. Thus, C(S) is, in fact, completely determined by its 1-skeleton. Higher-dimensional simplices do not contribute any new combinatorial information, but they make the homotopy theory of C(S) much more relevant and interesting. Speaking about the homotopy properties of C(S) we, of course, have in mind the homotopy properties of its geometric realization (cf. [216]), which is a topological space and even a CW-complex. Obviously, C(S) is an infinite complex. It is even locally infinite when it has positive dimension: every vertex is connected by an edge with an infinite number of vertices. But it is finitely-dimensional: its dimension is equal to the maximal number of pairwise nonisotopic disjoint non-trivial circles on S minus 1. It follows that if 5 is a connected orientable surface of genus g with b boundary components, then the dimension of C(S) is equal io3g — 4-\-b, except in the case g =1 and b = 0, when the dimension is zero (if this formula leads to a negative number, the complex of curves is empty). 3.2. Connectivity of C(S) and the Morse-Cerf theory Before discussing other homotopy properties of C{S), we will present a proof of the connectedness of C(S). This proof can be generalized to give the best possible connectivity re-

Mapping class groups

543

suit for closed surfaces S and surfaces S with one boundary component; cf. Theorem 3.2.C (the case of surfaces with more boundary components can be reduced to the case of closed surfaces in at least two different ways; cf. 3.3). It is based on some elementary facts from the theory of singularities, more precisely, on a generalization of the Morse theory to famines of functions due to Cerf [32,33]. Namely, we need the following result. LEMMA 3.2. A. Any family of functions {f'-S-^ R}o^/^i can be approximated (say in the C^-topology) by a family offunctions {gt}o^t^\ such that all functions gt belong to one of the following three classes: (i) Morse functions with all critical values {i.e., values at critical points) different, (ii) Morse functions with two equal critical values and all other critical values different from them and from each other; (Hi) functions having all critical values different and exactly one non-Morse critical point such that in appropriate local coordinates (x, y) around this critical point the function has the form x^ ± y^ + cfor some constant c G R. This lemma was originally proved in [32, Chapter II], and has become well known since then. The proof shows that one can also require that all but finitely many functions gt belong to the first class, but we do not need this fact. For a general introduction to the relevant ideas of the theory of singularities one can recommend a nice short book by Poston and Stewart [204]; cf. [204, Chapters 1,2]. THEOREM 3.2.B. Suppose that S is neither a sphere with ^ 4 holes, nor a torus with ^ 1 holes. Then C(S) is connected. PROOF. If 5 is a torus with two holes, it is easy to see directly that C(S) is connected. We leave this as an exercise to the reader, with the following hints: there are two types of non-trivial circles, namely, nonseparating circles and circles separating S into a torus with one hole and a disc with two holes; the circles of the first type are never disjoint, but two such circles intersecting at exactly one point can be made (by an isotopy) disjoint from a circle of the second type. In the remaining part of the proof we assume that S is neither a sphere with ^ 4 holes, nor a torus with ^ 2 holes. Let y = (D), y^ = {D') be two vertices of C(S). We need to prove that there is a sequence of vertices y = yo. • • •. KA? = K^ such that yi is connected by an edge with yz+i (or is equal to yi+x) for all / = 0 , . . . , n — 1. To begin with, let us choose two smooth functions f,f^:S->R such that D (respectively Z)0 is a component of a level set f~^(a) of / (respectively, of f) and, moreover, / and /^ have no critical points on D and D\ respectively. In addition, we may assume that both / and f are Morse functions having different critical values. Clearly, we can connect / and f by a path {//: 5 ^- R}, 0 ^ / ^ 1, in the space of smooth functions,

/ = /o, f = f\' Let {gr }o^?^i be some approximation of {//}o^?
544

N. V Ivanov

C' of some level set of g\ belongs to the isotopy class y'. The following claim provides similar components of level sets for all functions gt • CLAIM. If a function g\S ^^Vi belongs to one of the three classes of Lemma 3.2. A, then some component of some level set g~^{r) of g contains no critical points and is a nontrivial circle.

In order to prove this claim, let us choose a critical point x of g which is not a local maximum or minimum (such a critical value exists, because S is not a sphere or a disc). Let c = g(x) and let L be the component of the level set g~\c) containing x. Choose an ^ > 0 such that c is the only critical value in [c — 8,c -\- e]. Then exactly one component of the set g~\[c — 6, c -\- e]) contains L. Let us denote this component by Lg. If one of the boundary components of Lg is non-trivial, then we are done. Suppose that all of them are trivial, i.e., bound in S either a disc, or an annulus together with a boundary component of S. If one of these discs or annuli contains L^, then we replace g by a function g^ equal to g outside this disc (or annulus) X and having only one Morse critical point on it if it is a disc (or no critical points if it is an annulus). We may assume that if g^ has a critical point in X (i.e., if X is a disc), then the corresponding critical value is different from all other critical values of g^ Note that if Z is a disc, then g has at least two critical points on X (the point X and a minimum or a maximum), and g' has only one. If X is an annulus, then g has at least one critical point on X (namely, the point x), and g^ has none. In both cases g' has fewer critical points than g. If some component of some level set of g^ is non-trivial, then the same is true for g itself, because all components of level sets of g^ contained in X are trivial. So, in this case our problem is reduced to a similar problem for the function g\ which has fewer critical points than g. Since g' obviously belongs to one of three classes of Lemma 3.2.A, we can use induction. If none of these discs or annuli contain L^, then S is equal to the union of Ls with these discs or annuli. It follows that the Euler characteristic x (S) is equal to x i^s) + d, where d ^ 0 is the number of the discs. Clearly, L is a deformation retract of Lg and, hence, X(S) = X (L) + d. But, X (L) = — 1 if ^ contains one Morse critical point, x (^) = —2 if L contains two Morse critical points and x (^) = 0 if L contains a non-Morse critical point (in the last case L does not contain any other critical points and is a topological circle with a cusp at the critical point). The various possibilities for L are presented in Figure 2. It follows that X (S) ^ —2 in this case. But any such S is either excluded by the assumptions of the theorem, or is a torus with two holes, which is also excluded now. This completes the proof of the claim. Now we return to the proof of the theorem. The above claim implies that for any t, 0 ^ t ^ 1, the function gt has a component Ct of a level set g^ {at) such that Ct does not contain any critical point of gt and is a non-trivial circle. We may assume that Co = C and Ci = C^ If M is sufficiently close to t, say u belongs to a neighborhood Ut of ^ then the level set g~\at) of the function g^ has a component Ctu close and isotopic to C/, because gt has no critical points on Ct. The family of neighborhoods {L^rlo^r^i forms an open cover of the interval [0, 1]. By using a well known Lebesgue lemma we can divide the interval [0, 1] by several points 0 = xo^JCi ^ • • • ^ x „ = l into intervals [x/, jcz+i]

Mapping class groups

(a)

(b)

545

(c) Fig. 2.

such that any interval [JC/, jc/+i], 0 ^ / ^n — 1, is contained in a neighborhood [//.. We may assume that ^o = 0 and tn-\ =1. Let Yi = {Ctj). We claim that yo, • • •, y« is the required sequence of vertices. By our choice, yo = y and y^ = y\ Let us prove that yt is connected by an edge with y/+i for i =0,.. .,n — l. Let 0 ^ / < « — 1, and let x = x/, y = ^/+i, z = x/+2 and v = ti,w = ^/+i. Note that Ci, is isotopic to Cyy because y e[x,y]cUy and Cw is isotopic to Cwy because y e [y,z] C U^,. Hence, y,- = (C,,) = (C,) = (Cyy) and )//+i = (C,,^,> = (C^) = (C^y). But, Ci;jy; and Cwy are components of the level sets of the same function g^;. Hence, they are either equal or disjoint and their isotopy classes y/ and y+i are either equal or connected by an edge. This completes the proof. D This theorem for closed surfaces S was originally deduced by the author [96] from the results of Hatcher and Thurston [91], which were also based on the ideas of the Morse-Cerf theory. Later on, the method was streamlined and extended to cover the higher connectivity of C(S); cf. the discussion below. In the meantime, Harer [77,78] proved that C(S) is homotopy equivalent to a bouquet of spheres, all having the same dimension, and computed this dimension, giving, in particular, the best possible estimate of the higher-connectedness. Cf. 3.3. In [32] Cerf also proved a version of Lemma 3.2.A for 2-parameter famihes of functions. Here, some new types of functions appear, such as Morse functions with three equal critical values or functions having critical points of the form ibx"^ ± y^ -\- c. Using this version of Lemma 3.2.A, one can modify the proof of Theorem 3.2.B and prove the simply connectivity of C(S) in most cases. Instead of a subdivision of the interval [0, 1] into subintervals, one should use a sufficiently fine triangulation of a disc. The Morse-Cerf theory is based on a complete classification of possible singularities of functions in generic families. Such a classification is unknown for families with sufficiently many parameters. But, in fact, one can carry out the main arguments of the proof of Theorem 3.2.B without complete classification. One needs to prove only some restrictions on the possible complexity of singularities of functions in generic families (cf. [98, Lemmas 2.3 and 2.4]). This program is carried out in [98], Sections 1 and 2 and leads to the following result. THEOREM 3.2.C. If S is a closed surface (respectively, a surface 1 boundary component, a surface with > 2 boundary components), then the geometric realization of C(S) is —x(5) — I =2g — 3-connected (respectively, —x(S) — 2-connected, —x(^) — 3connected), where x(^) ^^ ^he Euler characteristic of S.

546

A^. V Ivanov

This result is not the best possible for surfaces with ^ 2 boundary components (the best is the — X (^S") — 2-connectedness), but the closed case is sufficient for the main appHcations (cf., for example, 6.4), and, in fact, the general case can be reduced to it. 3.3. The homotopy type of the complexes of curves The following theorem gives an almost complete description of the homotopy type of C (5). THEOREM 3.3.A. The geometric realization of C(S) is homotopy equivalent to a bouquet of spheres of dimension —x(S) if S is closed and of dimension —x(S) — I if S has nonempty boundary, where x(S) is the Euler characteristic ofS.

This theorem is due to Harer [78]. His proof is based on a connection of the complexes of curves with ideal triangulations of Teichmiiller spaces (cf. 5.5) and a combinatorial argument used to give an upper estimate of the homotopy dimension of C(S), i.e., of the minimal dimension of a CW-complex homotopy equivalent to it. This upper estimate, in fact, matches the connectivity results, and together they imply that C(S) is homotopy equivalent to a bouquet of spheres. Another proof, based on Theorem 3.2.C, a duality between the properties of C(S) and Mod^ (cf. 6.1 and 5.4) and Harer's combinatorial argument giving an upper estimate of homotopy dimension was suggested by the author [98]. We refer to [98] for a detailed presentation of this proof and a discussion of its relations with Harer's approach; cf. [98, Section 6]. Both approaches use an induction on the number of boundary components reducing, in fact, the general case to the closed one. Harer's inductive argument [78] is made on the geometric level of complexes of curves. He shows that making a hole in a surface with nonempty boundary increases the connectivity of C(S) by 1 (making the first hole is more complicated). In [98], this inductive argument is made on the algebraic level of the mapping class groups, where the exact sequences of 2.8 together with theory of groups with duality [13] provide a convenient tool. An improved version of this argument is used in the proof of Theorem 6.4.B. Table 1, summarizing the dependence of the homotopy dimension hdimC(5') and the usual dimension dimC(5) of C(S) on the genus g and the number b of boundary components of 5, is often quite useful. Table 1 g

b

hdim C(S)

8=0 g=o 8=1 8=^ 8^2 8^2

b^3 b^3 b=0 b^\ b=0 b^ 1

-1 b-4 0 b-l 28-2 2g-3 + b

dime (5) -1 b-4 0 b-l

3g-4 3g-4-i-b

COROLLARY 3.3 .B . Suppose that S is connected and has nonempty boundary. Then the structure of the simplicial complex C{S) (considered up to isomorphism) determines if the

Mapping class groups

genus of S is ^l,

547

and the whole topological type of S if it turns out that the genus of S is

The table shows that hdim C(5) = dim C(5) if and only if the genus is ^ 1 or the genus is 2 and the boundary is empty. The last line of the table allows us to find g and b if we know already that g^2 and b^l. D PROOF.

3.4. Hyperbolic properties Now we turn from the homotopy properties to the geometry of complexes of curves. Let C\ (S) be the 1-skeleton of C(S), i.e., the graph with the same vertices and edges as C(S). As we noted in 3.1, C(S) is a flag complex and, hence, C\ (S) completely determines the whole structure of C(S). Any graph or, rather, its geometric realization, has a canonical structure of a metric space, which is constructed as follows: any edge is made isometric to the interval [0,1] and then the distance between two points is defined to be the infimum of the lengths of paths connecting them. Clearly, the distance between two vertices y, y^ is equal to the length n of the shortest chain y = yo, • • •, K« = K^ such that y/ is connected by an edge with y/+i (compare the proof of Theorem 3.2.B). This structure of a metric space turns any graph into a geodesic space. This means that any two points x, y can be connected by an isometric image of an interval in the real line R. Any such image is called a geodesic and is denoted by [xy], despite the fact that it is usually not unique. A geodesic metric space X is called 8-hyperbolic, where 8 ^ 0 is some real number, if for any three points x,y,z e X any geodesic [xz] is contained in a 5-neighborhood of the union [xy] U [yz]> A geodesic metric space is called hyperbolic if it is 5-hyperbohc for some 8^0. These notions are due to Gromov [67] and serve as the starting point of his theory of hyperbolic spaces and groups. The main examples are provided by the classical hyperbolic spaces, (infinite) trees (i.e., graphs without cycles), and Cay ley graphs of fundamental groups of closed negatively curved manifolds. (For trees one can take 8 = 0.) Note also that any geodesic metric space of finite diameter D is trivially 5-hyperbolic with 8 = D. THEOREM 3.4.A. If C\(S) is connected, then its geometric realization is a hyperbolic space of infinite diameter

This remarkable and completely unexpected theorem was recently proved by Masur and Minsky [156]. An overview of the proof is provided by Minsky [165]. The replacement of C{S) by C\{S) is not significant: we can turn the geometric reahzation of C{S) into a metric space by making every simplex be isometric to a regular simplex with edges of length 1. Then the resulting metric space will also be hyperbolic. This easily follows from Theorem 3.4.A and basic properties of hyperbolic spaces. It simply seems that it is more convenient to work with C\{S) in this context. Even the fact that C\ (S) is of infinite diameter is non-trivial and interesting. It means that for any natural number N there are two vertices of C(S) which cannot be connected

548

N.V.Ivanov

by a chain of edges shorter than A^. This seems to be nearly obvious, but, apparently, was not proved before [156]. The fact that C{S) is not locally finite presents a serious obstacle if one attempts to use Theorem 3.4.A along the fines of Gromov's theory of hyperbolic groups and spaces [67]. In [157], Masur and Minsky developed some tools to overcome this obstacle. As a first application to the mapping class groups, they proved the following theorem about pseudoAnosov elements (for the definition, see 7.1) of the mapping class groups. 3.4.B. Let us fix a finite set of generators of Mods cii^d let us denote by \'\\^ the minimal word length with respect to these generators. If f\, fi cire two conjugate pseudo-Anosov elements of Mod^, then f\ = gf2g~^ for an element g e Mods such that THEOREM

lglw^C(|/,k + l/2k), where the constant C depends only on S and the generating set. To put this theorem into proper perspective, one should note that Hemion [92] proved that the conjugacy problem for the mapping class group is (algorithmically) solvable. Pseudo-Anosov elements are, in a definite sense, typical elements of Mod^, and Mosher [180] gave an explicit algorithm for determining the conjugacy for pseudo-Anosov elements. In both [92] and [180] no explicit bounds on the time required by the algorithm was given. Theorem 3.4.B obviously implies an explicit bound for the conjugacy problem, because it provides a bound on the word length of a conjugating element.

4. Dehn twists, generators, and relations The main examples of nontrivial elements of mapping class groups, and also the main building blocks in various constructions of other elements, are the so-called Dehn twists. They were introduced for the first time by Dehn [44], and then played a central role in his fundamental paper [45]. This section starts with a definition and some basic properties of Dehn twists, and then proceeds to one of their main applications, namely, to their role as generators of mapping class groups. The material of this section is crucial for developing an intuitive feeling of the mapping class groups. In this section S is always assumed to be a compact oriented surface.

4.1. Dehn twists The goal of this section is to define Dehn twists and to prove their basic properties. Dehn twists are, by definition, the isotopy classes of some special diffeomorphisms called twist diffeomorphisms. We begin with a discussion of the latter. We start with a description of a standard twist diffeomorphism of an annulus. Let A be the annulus in R^ given by the inequality 1 < r < 2 in the standard polar coordinates (r, 0) in R^. Its boundary 9A consists of two components 9i A, 82A defined by the equations r = 1, r = 2 respectively. Let us fix a smooth function <^ : R ^- R such that (p(x) =0 for

Mapping class groups

549

Fig. 3.

jc ^ 1, (p(x) = 27r for X > 2, 0 ^ (^ ^ lit, and the derivative (p^ is > 0. Let us define a diffeomorphism 7 : A —> A by the formula r(r, ^) = (r, ^ + <^(r)) in our polar coordinates (r, ^). We call T the standard twist diffeomorphism of the annulus A. It is easy to see that, up to an isotopy fixed on the boundary 9A, the diffeomorphism T does not depend on the choice of (p. Figure 3 illustrates the action of T. In fact, up to an isotopy fixed on the boundary, T is the unique diffeomorphism fixed on the boundary and taking the radial arc / in Figure 3(a) into an arc isotopic to the arc in Figure 3(b). (Using the fact that Diff(/ fix dl) is connected, any two such diffeomorphisms can be made equal on / by an isotopy. By an additional small isotopy, these two diffeomorphisms can be made equal on a neighborhood A^ of / U 9 A. Consider a disc D contained in A with the boundary contained in N. Since Diff(Z)fix9) is connected by Theorem 2.7.C, any two such diffeomorphisms can be made equal by an isotopy.) So, Figure 3 defines T up to an isotopy fixed on 9A. Now, let ^: A ^ 5 be an orientation-preserving embedding of A into our surface S. We may use e to transplant T from A to 5 as follows: take the diffeomorphism e o T o e~^ : e(A) -^ e(A) and extend it by the identity to a diffeomorphism Te'.S-^S. We call any such diffeomorphism Te a twist diffeomorphism of S. Clearly, up to isotopy fixed on 95, the diffeomorphism Te depends only on the isotopy class of the embedding e. The isotopy class of an embedding e:A-^Sis determined by the isotopy class of the (oriented) image e(a) of the oriented (say, counterclockwise) axis a = {(r,0): r = 3/2} of the annulus A and by which boundary component of A is mapped to the right side of e(a) (the notion of the right side is determined by the orientations of e(a) and S). So, given the isotopy class of the unoriented image e{a), there are four possible isotopy classes of e, but only two of them are orientation-preserving. In fact, if e is orientation-preserving, then e has to map 92A to the right of ^(a). It is easy to see (draw a picture!) that both orientation-preserving embeddings e with the same isotopy class of e{a) lead to isotopic diffeomorphisms Te. In particular, up to isotopy fixed on 95, the diffeomorphism Te depends only on the image C = e(a) of the axis a of A, and even only on its isotopy class y = (C). By a slight abuse of the language, we call this diffeomorphism a twist diffeomorphism about C. Clearly, any twist diffeomorphism preserves orientation and is fixed on the boundary of S. We call the isotopy class of a twist diffeomorphism about a circle C the Dehn twist about C. Since this isotopy class depends only on the isotopy class y = (C) of C, we usually denote it by ty and call it also the Dehn twist about y. There is an ambiguity in these notations and terminology; namely, one may consider only isotopies fixed on the boundary and then ty e Ms, or one may consider all isotopies and then ty e PMod^. The context usually resolves this ambiguity.

550

N.VIvanov

The Dehn twists we defined are often called the left Dehn twists, and the elements of the form t~' are called the right Dehn twists. Similarly, one may speak about the left and the right twist diffeomorphisms. This terminology is justified, first, by the fact that the right twist diffeomorphisms and Dehn twists can be constructed in a manner completely similar to the construction of the left ones, but using orientation-reversing embeddings A-^ S instead of orientation-preserving ones (we leave this as an exercise). Second, one can distinguish between the right and the left twist diffeomorphisms in the following way. Let ^: A ^^ 5 be an embedding and Te be the corresponding (left or right) twist diffeomorphism. If J is an arc in S transversely intersecting C\ = e{d\A), then as we approach the annulus e(A) along Te(J) and pass the circle Ci, we will be turning left if Te is a left twist diffeomorphism and we will be turning right if Te is a right twist diffeomorphism. Of course, the meaning of turning left or right is derived from the orientation of S. A useful exercise is to convince yourself that we will turn to the same direction if we approach e(A) from the other side crossing C2 = e(d2A). Curiously enough, Dehn himself [44,45] did not distinguished between left and right twists in the notations (or otherwise), but always described explicitly which twist he deals with. Let us consider now twist diffeomorphisms about trivial circles. lfe(a) bounds a disc D in 5, we may assume that D contains d\ A (and not 62 A), replacing, if necessary, ^ by ^ o /, where i: A^^ Ais defined by the formula i (r, 0) = (3 — r, In — 0) (clearly, e(a) = eoi(a), but e and e o / belong to different isotopy classes). Then we can extend e to an embedding ^':D2 ^ 5 of the disc D2 = {(r,0): r ^ 2} such that e\D3/2) = D, where D3/2 = {(r,0): r < 3/2}. We can use e^ to transplant the isotopy {T^: D2 -^ ^2}O^M^I of D2 defined by the formula T^(r, 0) = (r,0-\- (p(r) + U(27T - (p(r))) from D2 to S in the same manner as we transplanted T from A to S. Clearly, the isotopy {TJIO^M^I is fixed on the boundary d 02 = 82 A and connects the extension TQ of T to D2 by the identity with r [ = ido2 (note that (r,0) = (r,0-\-27T)). Hence, our transplanted isotopy connects Te with the identity diffeomorphism of S and is fixed on dS. Therefore, any twist diffeomorphism about a circle bounding a disc in S is isotopic to the identity by an isotopy fixed on dS. A similar argument shows that any twist diffeomorphism about a circle C bounding an annulus together with a boundary component of S (or equal to a such component) is isotopic to the identity; this time by an isotopy moving this boundary component. It follows that, in PMod^, any Dehn twist about a trivial circle (i.e., a circle bounding a disc or homotopic to a boundary component) is trivial (i.e., equal to 1). But in A^^ only the Dehn twists about circles bounding a disc are trivial and the Dehn twists about circles homotopic to a boundary component are nontrivial (of course, the last statement requires a proof). Now, we prove some basic properties of Dehn twists. We start with a refinement of the second statement of Corollary 2.7.E. LEMMA 4.1 .A. The group 7ro(Diff(A fix dA)) is an infinite cyclic group generated by the isotopy class of the standard twist diffeomorphism T. Consider the fibration Diff(D2fixa) -> Emb(Z)i, 1)2) from Theorem 2.6.A (recall that Di =[{r,0): r ^ /}; / = 1,2). Since Diff(D2fixa) is contractible (by Theorem 2.7.C) and since Emb(Di, D2) is weakly homotopy equivalent to the unit tangent PROOF.

Mapping class groups

551

bundle of D^ (by Theorem 2.6.C) and hence to a circle, the homotopy sequence of our fibration implies that 7ro(Diff(A fix 9 A)) is isomorphic to Z. Obviously, K\ (Emh(D\, D2)) is generated by the homotopy class of the loop uniformly rotating D\ once around its center. To compute the image of this generator in 7ro(Diff(A fix 9A)) we need to extend this rotation to an isotopy of D2 fixed on dD2. This was done, in fact, in our discussion of twist diffeomorphisms about trivial circles above. The extended isotopy ends in a diffeomorphism equal to the identity on D\ and to the standard twist diffeomorphism on A. The lemma follows. D COROLLARY 4.I.B. Let C be circle on S, and let H,H':S ^ S be two orientationpreserving diffeomorphisms leaving C invariant and preserving the orientation ofC. If the results of cutting H and H' along C are isotopic {with free boundary), then, up to isotopy, H differs from H' by a power of a twist diffeomorphism about C.

Changing H and H\ if necessary, by an isotopy, we may assume that both H and H^ are equal to the identity on C (because Diff(C) is connected). Obviously, both H and H' preserve the sides of C, and hence H and H' may be assumed to be equal to the identity on some annulus X having C as one of the boundary components. If we remove the interior of X from 5, we get a surface Sx diffeomorphic to the result Sc of cutting S along C. Obviously, the diffeomorphisms Hx, H'^ 'Sx ^^ Sx induced by H, H' are isotopic (after a natural identification of Sx with Sc) to the results of cutting //, H' along C. Hence, Hx, H'^ are isotopic. By extending this isotopy to 5, we get an isotopy between H and some diffeomorphism H'^ which is equal to H' outside X and hence is equal io H' oG, where G is some diffeomorphism supported^ in X. Since X is an annulus, an application of the lemma completes the proof. D PROOF.

LEMMA

4.1 . C . Iff^ ftyf

Mod^ and y is the isotopy class of a circle on S, then

=tf(y).

If ^: A ^- 5* is an embedding and F : 5 ^- 5 is a diffeomorphism, then, clearly, F oTe o F ~ ' = Tfoe- The lemma follows. D

PROOF.

COROLLARY

4.I.D. Any two Dehn twists about nonseparating circles on S are conju-

gate. PROOF.

It is sufficient to apply Lemma 2.3.A.

D

LEMMA 4.1 .E. Let C and D be two circles on S transversely intersecting at one point and let y, 8, respectively, be their isotopy classes. (i) tyts(y) = S.

(ii) tstytsiy) = y, and tgtyts can be represented by a diffeomorphism taking C to C and reversing the orientation ofC (or, what is the same, interchanging the sides ofC). The support of a diffeomorphism / is the closure of the set {x: f(x) ^ x}

552

N. V Ivanov

Fig. 4.

PROOF. First, notice that for any two pairs of circles transversely intersecting at one point there is a diffeomorphism of S taking one pair to the other. So, it is sufficient to prove the lemma for one such pair; for example, for the pair C, D in Figure 4(a) (the orientations of circles play a role only in the proof of (ii)). (i) The image £* of C under a twist diffeomorphism about D is illustrated in Figure 4(b), together with a circle isotopic to C. The image of E under a twist diffeomorphism about this last circle is illustrated in Figure 4(c). Clearly, the circle in Figure 4(c) is isotopic to the circle in Figure 4(d), which is nothing other than D. This proves (i). (ii) In order to prove that tsthsiy) = y, it is sufficient to apply (i) twice. In order to prove the remaining part of (ii), one needs to trace the orientations of our circles. The image of C oriented as in Figure 4(a) under the composition of two twist diffeomorphisms from the proof of (i) is illustrated in Figure 4(d). This image is isotopic to D with the orientation given by turning counterclockwise the orientation of C at the (unique) intersection point of C and D. If we apply these two diffeomorphism once more, but in the other order (so we get an oriented representative of tstytsiy)), we get C with the orientation given by turning the last orientation of D again counterclockwise. Clearly, this orientation of C is opposite to the original one. This proves (ii). D

I learned Lemma 4.l.E from Birman's survey [17], but the assertion (i) is definitely due to Dehn [45] (cf. [45, §4a])), as are also Lemma 4.1 .C and Corollary 4.1 .D. Now we prove several fundamental relations between Dehn twists. LEMMA 4.1 .F. Let C and D be two circles on S and let y, 8, respectively, be their isotopy classes. (i) IfC r\D = 0, then tyts = t^ty.

Mapping class groups

553

(ii) If C and D intersect transversely at exactly one point, then the following Artin relation holds: tytsty

=tstytS.

PROOF, (i) If C n D = 0, then, obviously, ty and ts can be represented by twist diffeomorphisms with disjoint supports. Such diffeomorphisms obviously commute. (ii) tytsty = tstyts IS obvlousty equivalent to t^tyt^^ = t~hsty. By Lemma 4.1.C, the latter equation is equivalent to ^t8iy)=^t-'

(S)'

Hence, it is sufficient to prove that ts(y) = t ^ (8), or, what is the same, tytsiy) = 8. But this is the content of Lemma 4.1 .E(ii). D LEMMA 4.1 .G. Let C and D be two circles on S transversely intersecting at one point and let y, 8, respectively, be their isotopy classes. Let N be a regular neighborhood of the union CUD (this means that N cut along C L) D is an annulus with comers) and let B = dN, p = (B). Then (tytsf = t^. PROOF. Instead of A^ we may consider the standard torus with one hole resulting from the identification of the opposite sides of a square having a small hole around its center. As C and D we may take circles represented by a vertical and a horizontal segment in this square, respectively. Let V and H be the corresponding twist diffeomorphisms. Consider the horizontal arc h in Figure 5(a). Figures 5(a)-(d) compute H oV{h) (up to isotopy). If we apply V to the arc in Figure 5(d), we get the same arc. So, the arc in Figure 5(d) is isotopic to V o H o V(h). Outside the big circle in Figure 5(e) the arc in Figure 5(d) looks exactly as the arc in Figure 5(a) turned 90°. Such a turn preserves orientation and hence interchanges V and H (more precisely, conjugates one to the other). So, by turning Figures 5(a)-(d), we can find the image of the arc in Figure 5(d) under HoV oH, i.e., we can find (H o V)^(h) = (H o V o H) o (V o H o V)(h). The result is the arc in Figure 5(f). This arc coincides with h outside the big circle in Figure 5(g). This allows us to find the image of the arc in Figure 5(f) under (H oV)^ or, what is the same, (H o V)^(h). Clearly, the result is isotopic to the image of h under a twist diffeomorphism about the boundary (the small circle in our pictures). Next, we would like to find (H o V)^(v), where i; is a vertical arc similar to h; cf. Figure 5(h). Using a 90° turn as above, we see that it is sufficient to find (V o H)^(h). Note that the first if in (V o Hf acts on h trivially, so (V o Hf(h) = (V o H)^ o V(h) = V o(H o V)Hh). But V o(H o V)Hh) = (H o Vf(h), because in the computation of (H o Vf(h) the last H acts trivially (like the last V in the computation of V o ^ o V(h)). Hence, (V o H)^(h) = (H o Vf(h), and hence (V o Hf(h) is equal to the image of h under a twist diffeomorphism about the boundary. It follows that the image (H o V)^(v) is equal to the image of v under a twist diffeomorphism about the boundary.

554

A^. V Ivanov

O

V

H

isotopy

(a)

<)-

(e)

(h)

(f) Fig. 5.

It remains to notice that if we cut our torus with a hole along h and v, we get a disc (with comers), and hence a diffeomorphism of our torus (fixed on the boundary) is determined, up to an isotopy fixed on the boundary, by its action on h and v (recall that Diff(D fix 9D) is connected for any disc D). D This lemma is also due (in a sHghtly different form) to Dehn [45] (cf. [45, §6c]). 4.1 . H . Let SQ be a sphere with four holes. Let the boundary components of SQ be Co,... ,€3 and for 1 ^ / < 7 ^ 3, let Cij denote a circle encircling Ci and Cj as in Figure 6. Suppose that SQ is embedded in S. Let tt ^ M.s be the Dehn twist about Ci, LEMMA

Fig. 6.

Mapping class groups

555

0 ^ / ^ 3, and let tij e Ms be the Dehn twist about dj, 1 < / < 7 ^ 3. Then the following lantern relation holds in Ms {cind hence in Mod^): tot\ht3

=t\2t\3t23'

PROOF. Connect Co with Ci, C2, C3 by three disjoint arcs / i , I2, h- Clearly, if we cut 5Q along these three arcs, we get a disc (with comers). Hence (as in the proof of Lemma 4.1.G), a diffeomorphism of S^ fixed on 95Q is determined up to an isotopy fixed on dS^ by its action on / i , h, h- Therefore, in order to prove our relation it is sufficient to compute the action of some representatives of both sides on the arcs / i , h, h> Of course, we will take compositions of twist diffeomorphisms as our representatives, and then their action on the arcs can be computed in a straightforward manner, as in the proof of Lemma 4.I.E. One just needs to draw several pictures, and we leave this task to the reader. D

The relation of Lemma 4.1.H was discovered by Dehn [45] (cf. [45, §7g) 1)]) and much later rediscovered and popularized under the name lantern relation by Johnson [113]. Finally, we will prove a lemma borrowed from [17] (cf. [17], the discussion after Corollary 3.3). It is crucial for the induction step in some arguments using the induction on the number of boundary components. Cf. Section 4.2. LEMMA 4.1 .L Suppose that dS ^0. Let R be the result of gluing a disc D to one of the boundary components of S. Let P e7i\(R,x) be an element represented by an embedded loop b. Without any loss of generality we may assume that the base point x is contained in the interior of D and the loop b intersects D in a single arc containing x. Then we can choose an annulus B containing D and b in its interior and having boundary components isotopic to b. Let C\ be the component of dB lying at the left side of b, and let C2 be the component of dB lying at the right side ofb, where the meaning of left and right is determined by the direction of the loop b and the orientation of S. Then

where j : 7t\ (R, x) -^ PMods is the homomorphismfrom 2.8, and y\ = {C\), Y2 = (C2). j(P) can be obtained in the following manner (cf. 2.8). Consider the loop b as an isotopy of the base point x and extend this isotopy to an isotopy {Ft: R -^ /?}o^/^i, Fo = id/?. We may assume that the ending diffeomorphism F\ of this isotopy is fixed on D. Then it induces a diffeomorphism F :S ^^ S, and j(P) is equal to the isotopy class of F. Obviously, the isotopy {/v}o^/^i may be chosen to be fixed outside of the annulus B. Moreover, it can be chosen is such a way that F\ is equal to the composition of a left twist diffeomorphism supported in an annulus (contained in B) with the boundary C\ and a right twist diffeomorphism supported in an annulus (disjoint from the first one and also contained in B) with the boundary C2. One can see this by drawing a couple of simple pictures or, alternatively, by defining {/vlo^r^i by exphcit formulas (compare our discussion of twist diffeomorphisms about trivial circles). We leave this as an exercise to the reader PROOF.

556

N.V Ivanov

(the pictures can be found in [101, Section (6.1)]). Clearly, this description of F\ implies the lemma. D

4.2. The Dehn-Lickorish theorem The main result of this section is Theorem 4.2.D, known as the Dehn-Lickorish theorem. It provides an explicit finite set of Dehn twists generating Mod^ (for closed S). We also discuss some corollaries of this result. Let us start with a couple of simple lemmas. LEMMA 4.2. A. The Dehn twists about the longitude and the meridian of a closed torus S generate Mod^. PROOF. By the discussion after Theorem 2.9.A, Mod^ is isomorphic to SL2(Z). This isomorphism, of course, depends on the choice of an isomorphism 7T\{S) ^-7?. If the latter isomorphism takes (the elements represented by) the longitude and the meridian into the standard basis of T?, then, as it is easy to see, the Dehn twists about the longitude and the meridian correspond to matrices

(i I) ^"^ ' (-1 i) On the other hand, it is well known that these matrices generate SL2(Z). The lemma follows. D LEMMA 4.2.B. Suppose that S is a closed surface of genus ^ 2. Let y, y^ be two vertices of C(S) represented by nonseparating circles on S. Then there is a sequence y = y\,y2,... ,yn = y^ of vertices ofC(S) such that any two consecutive vertices yi, }//+i are connected by an edge in C(S) and, moreover, if they are represented by two disjoint circles Ct, Q+i, then the union C/ U C/+i does not separate S. PROOF. Since C(S) is connected, there is a sequence y = 5i, ^ 2 , . . . , 5^ = y^ of vertices of C(5) such that any two consecutive vertices 5/, 5/+1 are connected by an edge in C(S). It may happen that some 5/ is the isotopy class of a circle D/ separating S into two parts S\, 52- In this case 5/_i, 5/+i can be represented by two circles A - i , A + i , respectively, disjoint from D/. If A - i and A + i are contained in different parts of 5, then D/_i fl Z)/+i = 0 and we can delete 5/ from our sequence. Suppose now that both Z)/_i and D/+i are contained in the same part of S, say in ^i. Since S is closed and D/ is nontrivial, the genus of 52 is ^ 1 and hence 52 contains some nonseparating circle D •. In this case we replace 5/ by (D^). After repeating this procedure several times, we will get a new sequence y =81,82,.. .,Sm = y^ such that all 5/ are the isotopy classes of nonseparating circles (and any two consecutive vertices <5/, 5/4.1 are connected by an edge). Now, let 1 < / ^ m — 1 and let A , A + i be two disjoint circles representing 5/, 5/+i respectively. Suppose that D/ U D/+i separates 5 into two parts 5i, 52. Since 5 is closed, both these parts have genus ^ 1 (we assume that 5/ / 5/+i). It follows that S\ (as also 52) contains some nonseparating circle D-. Clearly, both D/ U D- and D- U D/+i do not separate 5.

Mapping class groups

557

Let us replace the part 5/, (5/+i of our sequence by 8i = (D/), 5- = (D^^, 5/+i = (D/+i). After repeating this procedure several times, we will get the required sequence. D THEOREM 4.2.C. If S is a compact surface of genus ^ 1, then PMod^ is generated by Dehn twists about nonseparating circles. PROOF. We use the double induction on the genus and the number of boundary components. The induction starts with the closed torus. In this case the theorem follows from Lemma 4.2. A. Suppose that 95 7^ 0. Let R be the result of gluing a disc to one of the boundary components of S. By the inductive assumption, PMod/? is generated by twists about nonseparating circles. Obviously, every circle C on /? is isotopic to a circle C contained in 5, and tc = p{tc'), where p : PMod^ -^ PMod/? is the homomorphism from 2.8. Moreover, if C is nonseparating, then C' is also nonseparating. It follows that in order to prove the theorem for S it is sufficient to prove that ker p is generated by Dehn twists about nonseparating circles. The latter follows from Lemma 4.1.1 and the fact that 7T\ (R) is generated by the homotopy classes of (embedded) nonseparating loops. This completes the induction step of the induction on the number of boundary components. Next, suppose that S is closed. Let / e Mod^. Let us choose a nonseparating circle C on S. Consider two vertices y = (C>, y^ = f(y) of C(S) and connect them by a sequence y = Ki, K2, • • •, Km = y^ as in Lemma 4.2.B. For every / = 1, 2 , . . . , m — 1 we can represent }//, )//+i by disjoint circles C/, C/+i. Since C/ U C/+i does not separate S, we can choose a circle A transversely intersecting each of C/ and C/+i at exactly one point. Let 8i = (Di). By Lemma 4.1.E(i), /y,+,/5,(K/+I) = Si and t8ity.(8i) = yt. Hence, g{y') = y and g o f(y) = y, where g is the product of elements tSity-ty._^_^tSi, / = 1, 2 , . . . , m — 1. Since g o / ( y ) = y, we can represent g o / by a diffeomorphism H : S ^^ S preserving C. If H interchanges the sides of C, we replace g by ts^ ty^ ts^ g. Lemma 4.1 .E(ii) implies that for the new g the element g o f can be represented by a diffeomorphism H:S -^ S preserving C and preserving the sides of C. Now, let us cut S and H along C We will get a diffeomorphism He : Sc ^^ Sc preserving both boundary components of Sc • Hence, the isotopy class he of He belongs to PMods,^. Since the genus of Sc is less than the genus of S, he can be presented, by the inductive assumption, as a product of Dehn twists about nonseparating circles on Se • Every circle on Se is at the same time a circle on 5, and hence these Dehn twists in PMod^^ naturally define Dehn twists in PMod^. The product of these Dehn twists in PMod^ can be represented by a diffeomorphism H^ :S -^ S preserving C and isotopic to He after cutting along C. It follows that H differs from H^ (up to isotopy) by some power of a Dehn twist about C (cf. Corollary 4.1.B). Hence, g o / is equal to a product of Dehn twists about nonseparating circles. Since g is also such a product, by its construction, we conclude that / is a product of Dehn twists about nonseparating circles. This completes the induction step of the induction on genus, and hence the proof. D

This theorem is due to Dehn [45]. His work was forgotten for a while, and the theorem was rediscovered by Lickorish [132]. To a certain extent, our proof follows the proof of Birman [17] (cf. [17, Theorem 4.1]), who simplified the proof of Lickorish. The use of the connectedness of C(S) leads to further radical simplifications, and is, in fact, closer in

558

Fig. 7.

spirit to Dehn's original approach than to the one of Lickorish. The main tool of Dehn [45] was the action of Mod^ on the set of (the isotopy classes of) systems of circles. What was lacking in [45] (and is crucial to our approach) is an appropriate structure on this set, namely, the structure of a simplicial complex introduced much later by Harvey [87]. Now we are ready to prove the Dehn-Lickorish theorem. 4.2.D. If S is a closed surface of genus g, then Mods is generated by Dehn twists about the 3g — 1 circles A\,..., Ag, B\,..., Bg- \,C\,.. .,Cg presented in Figure 7. THEOREM

By Lemma 4.2.A the theorem holds for g = 1. Using induction on g, we may assume that the theorem holds for all surfaces of genus < g — 1. Let us denote by a i , . . . , a^, y^i,..., ^^_i, y i , . . . , y^ the isotopy classes of the circles A\,..., Ag, B\,..., Bg- \,C\,. ..,Cg respectively. Let us denote by Gs the subgroup of Mods generated by Dehn twists about these 3g — 1 circles. So, we have to prove that Gs = Mods- In view of Theorem 4.2.C it is sufficient to prove that Gs contains all Dehn twists about nonseparating circles. By its definition, the subgroup Gs contains some such twists. Hence, by Lemma 4.1.C it is sufficient to prove that Gs acts transitively on the set of isotopy classes of nonseparating circles. Moreover, since C\ is nonseparating, it is sufficient to prove that for every nonseparating circle C^ there is an element f e Gs such that fiyi) = y\ where y' = (C'}. Now we claim that it is sufficient to prove the last statement only for circles C^ such that Ci n C' = 0 and C\ U C^ does not separate S. In fact, suppose that our statement is proved for such special circles C^ and consider an arbitrary nonseparating circle C^ Let us connect y\ withy^ = (C^ by a sequence yi = yii, yi2, • • •, yi« = y^ as in Lemma 4.2.B. By our assumption, there is g G Gs such that g(yi) = g(yi i) = yn- Note that yi = yi i is connected with g~\y') by a shorter sequence y\ = y\\ = g~'(yi2), g"'(yi3), • • •, ^~kyin) = g~\y') with the properties of Lemma 4.2.B. Using induction, we may assume that g~^(y') = h(y\) for some h e Gs- Then g o h(y\) = y' and g oh e Gs. This proves our claim. So, let C^ be a circle such C\ fi C' = 0 and C\U C^ does not separate 5, and let y^ = {C'>. By Lemma 4.1.E(i) we have ty^ta^iyO =a\, ta^t^^{a\) = ^ i , tp^ta2(P\) = « 2 , and PROOF.

Mapping class groups

559

C2

Fig. 8.

^^2^2(^2) = K2- It follows that g(y\) = K2 for some g e Gs (namely, for the product of the above eight Dehn twists). Therefore, it is sufficient to prove that /(y2) = Y^ for some / G G5. Note that both C2 and C are disjoint from C\ and both C\ U C2 and C\ U C do not separate S. Let R be the result of cutting S along Ci, and let Q be the result of gluing two discs Di, D2 to the two boundary components of R. Clearly, both C2 and C are contained in Q (and even in R) and do not separate Q. Consider the subgroup GQ of Modg generated by Dehn twists about the circles A2,..., Ag, B2,..., Bg-\,C2, • ',Cg (clearly, all these circles are contained in Q). Cf. Figure 8. Since the genus of Q is equal to g — 1, we have GQ= ModQ by the inductive assumption. Since GQ = Modg, we can find a diffeomorphism F: Q ^^ Q such that F(C2) is isotopic to C^ in Q and the isotopy class of F belongs to GQ. Since Gg is generated by the Dehn twists about the circles A2,..., Ag, B2,..., Bg-\, C 2 , . . . , Cg, we may assume that F is equal to a composition of twist diffeomorphisms supported in some annuli with boundary components isotopic to these circles. Clearly, we may assume these annuli to be disjoint from the discs D\, D2 added to /?. In particular, we may assume that F is fixed on D\ and D2. Such a diffeomorphism F can be obtained by cutting a (unique) diffeomorphism F^ :S ^^ S fixed on C\ along C\ and gluing in the identity diffeomorphisms ido,, id/)2 • Obviously, F ^ is equal to a composition of twist diffeomorphisms supported in the same annuli as F. In particular, its isotopy class f^ belongs to GsIf F(C2) is not only isotopic, but equal to C^ then F^{C2) is also equal to C^ and hence f^{y2) = y' - This completes the proof in the case F(C2) = C^ since f^ ^GQ. In the rest of the proof we deal with the complications arising from the fact that in general we only know that F(C2) is isotopic to C^ in Q. Let {Kt: Q -> 2}o^r^i be an isotopy moving F(C2) into C^ so that ^0 = idg, ^1 (^(^2)) = C^ During this isotopy, the discs D\, D2 may be moved. Note that D\ and D2 are disjoint from F(C2) (because F is fixed on D\ and D2), and hence K\(D\) and K\ (D2) are disjoint from C. Since C^ does not separate Q, we can move the discs K\ (D\), K\(D2) back to their original position D\, D2 without moving C^ More precisely, there is an isotopy {L/: Q -^ Q}o^t^\ such that LQ = idg, Lt is fixed on C^ for all t, and L\oKi\Di =idD,, Li o K^ \"D2 =idD2-L^t H = L^ o/^,. Clearly, H{F(C2)) = C\

560

Fig. 9.

H is isotopic to 'idq and H \ D\ = ido,, H \ D2 = id/)2- Like F, the diffeomorphism H can be obtained by cutting a (unique) diffeomorphism H^ : S ^^ S fixed on C\ along Ci and gluing in the identity diffeomorphisms id^i, idD2- Let h^ be the isotopy class of H^. Since H o F(C2) = C\ we have H^ o F^(C2) = C and hence h^ o /^(y2) = /'• Recall that / ^ G G5 and hence h^ o f^ e Gs if h^ e G5. Therefore, in order to complete the proof it is sufficient to show that /z^ G G^. In view of the properties of H stated in the last paragraph, it is sufficient, in fact, to prove the following claim. CLAIM. Let H^ :S ^^ S be a diffeomorphism fixed on C\, preserving the sides ofC\ and such that the result H of cutting H^ along C\ and gluing in the identity diffeomorphisms ido,, idz)2 is isotopic to idg. Then the isotopy class h^ of H^ belongs to Gs.

Let us now prove this claim. First, by Lemma 4.1.E(ii) some representative I^ of the isotopy class i^ = ta\ ty^ ta^ maps C\ to Ci and interchanges the sides of Ci. Let F: R ^^ R be the result of cutting / ^ along C\ and let /' G Mod/? be its isotopy class. Next, by Lemma 4.1.G, t^ = (ty^ t^^ )^ is equal to ^5^, where 5o is the isotopy class of the circle Do in Figure 9. In particular, t^ can be represented by a twist diffeomorphism T^ about DQ. We may assume that T^ is supported in an annulus disjoint from C\. Let T' .R^^ /? be the result of cutting T^ along C\ and let t' G Mod/? be its isotopy class. Let G be the subgroup of Mod/? generatedby/^^^ and the Dehn twists about A 2 , . . . , A^, ^2, • • •, ^ ^ - i . G2,..., Q . Let H^ '.R->Rht\ht result of cutting H along C (recall that R = Sc) and let h^ e Mod/? be its isotopy class. Suppose that h^ e G, i.e., that h^ can be represented as a product of i\ t' and the Dehn twists about A2,..., Ag, B2,..., ^g-\^ C2,..., Q on R. Then h^ differs from the corresponding product of /^, r^ and the Dehn twists about A 2 , . . . , A^, ^ 2 , . . . , ^ ^ - 1 , C2,..., Q (considered now on S) by some power of the Dehn twist ty^ about C\. Since i^ = ta^ ty^ta^ and t^ = (ty^tc^)^ belong to G5, it follows that in this case h^ eGs- Therefore, it is sufficient to prove that /z^ G G. By the assumption, h^ belongs to the kernel of the natural map p : PMod/? -^ PMod^ from 2.8. So, it is sufficient to prove that keryo c G. The kernel kerp is equal to the image of the canonical homomorphism j :PB2(Q) -> Mod/? from 2.8, and hence is generated by the images of the fundamental groups 7t\(Q\), 7T\ (Q2), where Qi is the result of gluing only one disc A to R (and / = 1,2).

Mapping class groups

561

(a)

(b)

(c)

(d)

Fig. 10.

Consider first the element a of 7ti{Q2) represented by the loop in Figure 10(a). In view of Lemma 4.1.1, its image in Mod/? is equal to t^ty^, where yS, y are the isotopy classes of circles B, C from Figure 10(b). Clearly, 5 , C are isotopic in Rio B\, C2 respectively, and

562

A^. V. Ivanov

hence this image is equal to t^^ t~^ e G. Let us consider next the element b of 7t\(Q2) represented by the loop in Figure 10(c) (a loop going round the hole). In view of Lemma 4.1.1, its image in Mod/? is equal to tst~\ where 5, ^ are the isotopy classes of circles D, E from Figure 10(d). Clearly, D is isotopic to DQ and £" is a trivial circle (homotopic to a boundary component). It follows that ts =18^^, ts = I, and hence tst~^ = tsQ = t^. Therefore, the image of b also belongs to G. Recall that fj(cr)f-^ = j(Ma)) for any / e Mod/?, a e B2(Q) (cf. 2.8). It follows that j(f^(cr)) e G if / G G, 7 (a) e G. In particular, if the image of some element c e TtiiQi) belongs to G and f e G, then the image of /*(c) also belongs to G (if / is not in PMod/?, i.e., if / interchanges the holes, then /*(c) e7T\(Q\), but actually we do not need this case). Now, Figures ll(a)-(g) shows how to produce new elements of n\(Q2) from the element a by applying consecutively various Dehn twists from G to a. Clearly, together with b these elements generate 7t\(Q2). It follows that the image of 7t\(Q2) in Mod/? is contained in G. It remains to show the same for the image of 7t\(Q\). This can be done by similar pictures. But it is easier to note that conjugation by the element i\ interchanging the holes of R, maps the image of iri (^2) into the image of TTI((Qi). It follows that (since i^ G G ) the image of 7T\(Q\) is also contained in G. And since the images of n\(Q\), n\(Q2) generate kerp by Corollary 2.8.B and Theorem 2.8.C, we conclude that kerp c G. As we already said, this completes the proof of the claim, and hence of the theorem. D Dehn [45] was the first to prove that the groups Mod^ are finitely generated and to provide an explicit finite set of generators for them. His set of generators consisted of 2g(g — 1) Dehn twists for a closed surface S of genus g ^ 3 and of 5 Dehn twists for a closed surface of genus 2. Much later, and independent of Dehn's work. Theorem 4.2.D was proved by Lickorish [133,137]. An exposition of the work of Lickorish was given by Birman [16]; see [16, Chapter 4]. Again, to a certain extent, our proof follows the simplified proof of Birman [17] (cf. [17, Theorem 4.7 and Corollary 4.8]), and the use of C(5) leads to further radical simplifications and is closer in spirit to Dehn's original approach. (We also added some details corresponding to the proof of Corollary 4.8 from [17], omitted in [17].) COROLLARY

4.2.E. Mod^ is finitely generated for any compact (orientable) surface S.

PROOF. Since PMod^ is of finite index in Mod^, it is sufficient to prove that the groups PMod^ are finitely generated. Let us glue discs to the boundary components of S one at a time. Using the second exact sequence of Theorem 2.8.C with n = \ (recall that PB\ ( 2 ) = TT1 ( 2 ) for any surface Q), the fact that the fundamental groups of compact surfaces are finitely generated, and the induction on the number of boundary components, we can reduce our assertion to the case of closed surfaces S. But for closed 5, Theorem 4.2.D provides an explicit finite system of generators. D COROLLARY 4.2.F. If S is a closed surface of genus g, then Mod^ is generated by Dehn twists about the 2g -\-1 circles A\,..., Ag, B\,..., Bg-\,C\,C2 presented in Figure 7. PROOF. This result is due to Humphries [93], and we refer to [93] for a sequence of pictures showing how to transform C/ into a circle isotopic to C/+2 by a sequence of 16

Mapping class groups

(a)

(b)

(c)

isotopy

(d)

tp2

Fig. ll(a)-(d).

563

564

N. V Ivanov

(e)

(f)

isotopy

(g)

etc.

Fig. ll(eHg).

Mapping class groups

565

twist diffeomorphisms about A/, Bi, C/+i, A/+i, and 5/+i. In view of Lemma 4.1.C, this implies that one can express the Dehn twist about C/+2 in terms of Dehn twists about C/, A/, 5/, C/+1, A/+1, and 5/+i. It follows that we can consecutively eliminate Dehn twists about C^, C^_ i , . . . , C3 from our list of generators. D This result is complemented by a theorem of Humphries [93] to the effect that 2g -\- I is actually the minimal number of Dehn twist generators of Mod^ if 5 is a closed surface of genus g. If we do not require the generators to be Dehn twists, then one needs, in fact, only two generators. Namely, the following theorem holds. THEOREM

4.2.G. If S is a closed surface of genus g, then Mod^ is generated by the two

elements ^y\ l^a\t^\ ^^2^^2 ' ' ' ^^g-\ ^^g '

t

t-^

where ofi, ^^i,..., of^, yi, y^-i, Kg are, respectively, the isotopy classes of the circles A\, B\ . . . , Ag, Ci, Cg-1, Cg presented in Figure 7. This nice result is due to Wajnryb [231]. He deduced it from the above Corollary 4.2.F by some ingenious computations. His proof also covers the case of surfaces with one boundary component (if we made a hole in the surface in Figure 7 at the right end of it, then the Dehn twists about the same circles may serve as generators). The point of this theorem is in providing the minimal number of generators (clearly, Mod^ is not cyclic, and hence cannot be generated by one element!). As Wajnryb explains in [231], it is quite easy to find a generating set consisting of three elements, and already Lickorish [134] noticed that four elements generate Mod^. While the property of being generated by two elements has no immediate important consequences, it traditionally attracts the attention of group theorists.

4.3. Finite presentations After we have proved that the groups Mod^ are finitely generated, it is only natural to ask if they are finitely presented (and to look for a finite presentation, if they are). In the spirit of our proof of the Dehn-Lickorish theorem (i.e.. Theorem 4.2.D), we will base our approach to this question upon the action of Mod^ on C{S). Such an approach is made possible by the following general theorem. 4.3.A. Suppose that a group G acts on a CW-complex X permuting its cells and suppose that: (i) X is simply connected (inparticular, X is connected); (ii) the isotropy group of every vertex ofX is finitely presented; (iii) the isotropy group of every edge ofX is finitely generated; (iv) the number of orbits of cells of dimension ^ 2 is finite. Then G is finitely presented. THEOREM

566

N.V Ivanov

An elegant proof of this theorem, based on the Bass-Serre theory [214], is given by Brown [27]; cf. [27, Theorem 4]. But it can also be easily proved directly. In the special case when G acts transitively on the set of vertices, such a proof is contained in the report of Laudenbach [130] about the paper of Hatcher and Thurston [91]; cf. [130, §2]. Implicitly this special case was proved and used already by Hatcher and Thurston [91]. Laudenbach's proof can be easily extended to the general case. Brown [27] gives a long list of precursors of Theorem 4.3.A, but the papers [91,130] apparently were unknown to him. It is worth adding to his hst also the results of Behr [8, Section 1], and of Kozsul [129, Chapter III, §2]. While Behr [8] states his sufficient condition for the finite presentabiUty in terms of actions on discrete metric spaces, Koszul [129] reworked it in the language of actions on graphs, and obtained, in fact, a necessary and sufficient condition. Actually, as it is clear from his definition^ of the homotopy of loops, Koszul [129] works not with graphs, but with flag complexes (cf. 3.1). Compared with [129], the main novelty of Theorem 4.3.A lies in the fact that locally infinite complexes are admitted. According to Harvey [87], the fact that the complexes of curves are not locally finite also was an (at least a psychological) obstacle to overcome. Note that all proofs of Theorem 4.3.A allow, in principle, to construct a finite presentation of G starting from finite presentations of the isotropy groups of vertices, finite sets of generators of the isotropy groups of edges, the combinatorial structure of X, and, of course, the action of G on X. This leads usually to a fairly complicated presentation, which one may try to simplify. For Mod^, such an approach was successfully realized by Wajnryb in his work [230] discussed below. The following two lemmas admit direct and elementary proofs. But we prefer to deduce them from Theorem 4.3.A and the well known fact that for any finitely presented group there is a finite CW-complex having this group as its fundamental group. Our proof of the first lemma was suggested by Koszul [129, §111.2, Remark 2]. LEMMA 4.3.B. Let l-^H-^G^^F^^ I be an exact sequence of groups. If H and F are finitely presented, then G is also finitely presented. PROOF. Let 7 be a finite CW-complex with n\ (F) = F, and let X be the universal cover of y. So, X is simply connected and F freely acts on X. This action induces an action of G on X. Since F acts on X freely, the isotropy groups of vertices and edges under the action of G are all isomorphic to the group //, which is finitely presented. Since Y is finite, the number of orbits of cells (under the action of either F or G) is finite. Hence we may apply Theorem 4.3.A and conclude that G is finitely presented. D LEMMA 4.3.C. Let H be a subgroup of finite index of a group G. Then H is finitely presented if and only if G is finitely presented. PROOF. Suppose first that G is finitely presented. Let F be a finite CW-complex with 7X\{Y) = G, and let X be the universal cover of Y. Then H acts freely on X and the number of the orbits of the cells is finite because Y — X/G is finite and the index [G : //] is finite. Theorem 4.3.A now implies that H is finitely presented. in [129, p. 37], on the second line (AQ, ...,ai, ai^\,...,

an) should be replaced by (AQ, ... ,ai, <2/+2.

...,an).

Mapping class groups

567

Suppose now that H is finitely presented. Let

geG

This intersection is actually finite, because [G : H] < oo, and hence H^ is of finite index in both G and H. By the already proved part of the lemma, H^ is finitely presented. Since H' is obviously normal in G, we have an exact sequence of groups 1 -^ //^ ^- G -> G/H^ -> 1. Since G/H^ is finite, it is finitely presented. Now Lemma 4.3.B implies that G is finitely presented. D THEOREM

4.3. D . Mod^ isfinitelypresented for any compact (orientable) surface S.

PROOF. In view of Lemma 4.3.C it is sufficient to prove that the groups PMod^ are finitely presented. In order to deal with FMod^ we will use double induction on the genus and the number of boundary components, as in the proof of Theorem 4.2.C. Suppose that dS ^0. Let R be the result of gluing a disc to one of the boundary components of S. By the inductive assumption PMod/? is finitely presented. The kernel of the natural homomorphism p : FMod^ -^ PMod/? from 2.8 is finitely presented (by Theorem 2.8.C this kernel is isomorphic to 7T\ (R) with few exceptions; these exceptions are easy to deal with directly), and hence Lemma 4.3.C imphes that PMod^ is also finitely presented. This completes the induction step of the induction on the number of boundary components. Next, suppose that S is closed. If 5 is a sphere, then PMod^ = 1. If 5 is a torus, then PMod^ is isomorphic to SL2 (Z). It is well known that the latter group is finitely presented (cf. [214], for example). So, we may assume that the genus of 5 is ^ 2. In this case we are going to apply Theorem 4.3.A to the action of PMod^ on the geometric realization of C(S). We need to verify the assumptions of Theorem 4.3.A. Theorem 3.2.C implies (i) (cf. also the discussion preceding Theorem 3.2.C). Let us now check (iii). Suppose that (Gi), (C2) are connected by an edge of C(S), and let G be the isotropy group of (the geometric realization of) this edge. We may assume that C\ n G2 = 0. Let Go be the subgroup of PMod5 consisting of the isotopy classes of diffeomorphisms F :S -^ S, such that F(Gi) = Gi, FiCj) = G2 and F preserves the sides of both circles Gi, G2. Clearly, Go is a subgroup of finite index in G (in fact, the index is ^ 4). Such diffeomorphisms F can be cut along C\ U G2, and this leads to a canonical homomorphism Go -> PModg, where Q is the result of cutting S along Gi U G2. Clearly, this homomorphism is surjective, its kernel is generated by the Dehn twists about C\ and G2 (the latter assertion follows from Corollary 4.1.B applied twice), and these Dehn twists commute (by Lemma 4.1.F(i)). Since S is closed, the genus of the components of Q is (strictly) less than the genus of S. So, the inductive assumption implies that PMod^ is finitely presented. By Lemma 4.3.B, Go is also finitely presented. Finally, by Lemma 4.3.C the isotropy group G is finitely presented. So, we proved an assertion even stronger than (iii). The proof of (ii) is similar and simpler. In order to check (iv), notice that the number of orbits of cells of dimension < 2 is nothing more than the number of arrangements of ^ 3 disjoint circles on S considered up to a

568

N.VIvanov

diffeomorphism. Clearly, this number is finite. So, we can apply Theorem 4.3.A and conclude that PMod^ is finitely presented. This completes the induction step of the induction on genus, and hence the proof. D Theorem 4.3.D was originally proved by McCool [161] by methods of the combinatorial group theory. His approach allows, in principle, computation of an explicit finite presentation of Mod^ for a given 5, but this turned out to be too difficult to be done for any S. Later on, Hatcher and Thurston [91] suggested a geometric approach to the finite presentability and to a construction of an explicit finite presentation. The above proof was suggested by the author in [98]. It has some features in common with [91], such as the use of families of functions in order to prove the simply connectivity of an appropriate complex (cf. the proof of Theorems 3.2.B and 3.2.C above), and the use of Theorem 4.3.A (only implicit in [91]). In some sense, this proof can be considered as an ultimate simplification of the proof of Hatcher and Thurston (at least for the time being). One more approach to this theorem is indicated in 5.4. The complex used in [91] is very compUcated, and Hatcher and Thurston did not achieve their goal of writing down an explicit finite presentation (that does not diminishes the significance of their exceptionally beautiful paper!). Their complex was somewhat simplified by Harer [75], and using this simplified complex Wajnryb [230] succeeded in writing down an explicit finite presentation for closed surfaces and for surfaces with one boundary component. The proof involves some extremely complicated computations. The reader should be warned that [230] contains some mistakes, even in the statements of the main theorems, corrected much later in [21]. Recently, Wajnryb [232] provided a new exposition of his work. In addition, [232] contains a new proof of the simply connectivity of the HatcherThurston complex. Let us briefly describe the Wajnryb presentation for closed surfaces. As generators he uses the Humphries generators of Corollary 4.2.F. Relations are the following. First, for any pair of disjoint circles from the list of Corollary 4.2.F, the Dehn twists about them are related by the commutation relation of Lemma 4.1.F(i). Second, for any pair of circles from this list intersecting (transversely) at one point, the Dehn twists about them are related by the Artin relation of Lemma 4.1.F(ii). There are three additional relations. In order to describe them, let us, as usual, denote the isotopy classes of the circles A\,..., Ag, B\,..., Bg-\, C\, ...,Cg in Figure 7 by a\,.. .,ag, P\,,.., Pg-\, y\, . . . , y^,, respectively. In order to describe the first relation, we need a circle D2 on S intersecting transversely A 2 at exactly one point, not intersecting any other circle in Figure 7, and not isotopic to C2 (up to isotopy D2 is characterized by these properties; it is situated similarly to C2, but is contained in the lower part of S). Let 82 = {D2). Then 82 = u(y2), where u = /c^^^/^i ^a\ ty t^^t^^ 1^2, as one may check straightforwardly. Hence, r^^ =utY2U~^. It turns out that (fy, ta^ t^^ )^ = tyJsj. and using our formulas for t^^ ^^^ "' ^^ ^^^ express this relation in terms of the Humphries generators. This is the first additional relation. The next relation is, essentially, the lantern relation of Lemma 4.1.H, expressed in terms of the Humphries generators. To write it down, Wajnryb chose an embedding of SQ in S such that the boundary circles Co, Ci, C2, C3 of S^ are mapped to the circles Ci, ^2, ^3, C3 of Figure 7. Then he expressed the Dehn twists about the images of the

Mapping class groups

569

-p Ci

Ai

Bi

A2 Fig. 12.

circles C12, C13, C23, and the Dehn twist about C3 in terms of the Humphries generators. After this he wrote the lantern relation in terms of these expressions. This is the second additional relation. The last relation is concerned with the so-called hyperelliptic involution of S. It is an orientation-preserving diffeomorphism H: S -> S of order 2 (i.e., H o H = ids). Up to isotopy, it is characterized by the following properties: it maps each of the 2g -\- \ circles C\, A \, B\, A2, B2,..., Bg-\, Ag,Cg to itself, and reverses the orientation of all these circles (and hence interchanges the components of the complement in S of the union of these 2g -h 1 circles). It can be represented by a 180° rotation around the horizontal axis of symmetry in Figure 12. The isotopy class h of H is also called the hyperelliptic involution. (Note that, as it is described here, the hyperelliptic involution depends on the choice of a collection of circles diffeomorphic to the one in Figure 7, and different choices lead to different, but conjugate, hyperelliptic involutions. For the purposes of writing down the Wajnryb presentation, we need only one such collection of circles.) Since H{Cg) = Cg and H is orientation-preserving, hty^h~^ = ty^ by Lemma 4.1.C, or, what is the same, hty^ = ty^h. The third additional relation is, essentially, this commutation relation. It turns out that/z = ty^fty^,where f = ^ , % _ , . ^ ^tp^ta^t^^ta^t^^ .. .r^^,_,^c^^. Clearly, h commutes with ty^ if and only if / does. Wajnryb expressed ty^ in terms of the Humphries generators, and then, using this expression and the above formula for / , he wrote the commutation relation fty^ = ty^f in terms of the Humphries generators. This is his third (and the last) additional relation. The Wajnryb presentation for surfaces with one boundary component is obtained from the Wajnryb presentation for closed surfaces by omitting the third additional relation. (If we made a hole in the surface on Figure 7 at the right end of it, the Dehn twist about the same circles serve as generators.) The above description of the Wajnryb presentations in geometrical terms is, essentially, borrowed from Birman's survey [18]. (Note that the relation (6) in [18] is incorrect as stated. One should replace in it the element G by the above expression for the hyperelliptic involution/z.) Now, let 5 be a closed surface of genus 2. In this case the second additional relation follows from the other relations. The first finite presentation of Mod^ in this case was dis-

570

N.VIvanov

covered by Bergau and Mennicke [12]. Their proof turned out to be incomplete, and their presentation was justified only about ten years later by Birman and Hilden [19]. Instead of the first two additional relations of the Wajnryb presentation, this presentation contains the following two relations: y^y\ ^a\ ^fi\ ^ai^Yl^

^^ 15

(the second one expresses the fact that the hyperelliptic involution has order 2). A special feature of this case is that the hyperelliptic involution h commutes with all 5 generators of Theorem 4.2.D or Corollary 4.2.F (known already to Dehn [45]), as is clear from the above description of h, and hence belongs to the center of Mod^. In fact, it generates the center in this case. On the contrary, if the genus is > 3, then h does not commutes with r^j, and, in fact, the center is trivial. Cf. 7.5. For surfaces S of genus 0 (i.e., a spheres with holes), Mod^ is closely related to the classical Artin braid groups. This allows to write down a presentation of Mod^ in this case. For the details, see [16, Chapter 4]. Recently, Gervais [60] used the Wajnryb presentation and induction on the number of the boundary components in order to derive an explicit (and very symmetric) presentation of Mod^ for surfaces S with an arbitrary number of boundary components. Finally, let us mention a related work of Gervais [59], where he derived some infinite presentations of mapping class groups, similar to the Wajnryb presentation, but having as generators all Dehn twists or all Dehn twists about nonseparating circles. His presentation with all Dehn twists as generators was recently simplified by Luo [146]. 5. Teichmiiller spaces Teichmiiller spaces provide another class of fundamental geometric objects on which mapping class groups act. Our approach to Teichmiiller spaces is closer in spirit to that of Fricke [55] than to the approach of Teichmiiller himself. In particular, we define the Teichmiiller spaces in terms of metrics of constant curvature, and not in terms of Riemann surfaces as Teichmiiller did. In fact, the work of Fricke precedes the work of Teichmiiller by about forty years. Probably, Fricke was also the first to define the mapping class groups. In our discussion we stress the connection of Teichmiiller spaces with complexes of curves (and especially the results of Harvey [86,87]; cf. 5.4) and other combinatorial objects, namely, the so-called ideal triangulations (cf. 5.5). As the first application, we will indicate another approach to the finite presentability of the mapping class groups (cf. Corollary 5.4.B). In the next section we will present a much deeper application of Teichmiiller spaces to the mapping class groups, namely, the computation of the virtual cohomological dimension of the mapping class groups (cf. Theorems 6.4.A, 6.4.B and 6.4.C). 5.1. Definitions Let us assume that our (compact orientable) surface S admits a Riemannian metric of constant curvature —1 with geodesic boundary. As is well known (see 2.5), such met-

Mapping class groups

571

rics exist if and only if S has negative Euler characteristic. Let us fix some number / > 0 and consider only metrics such that all components of the boundary dS have length /. The choice of / is largely irrelevant, but at a later point it will be convenient to have / small. Let us denote by Hs the set of such metrics. We can endow Hs with some convenient topology, say the C^-topology, by considering Riemannian metrics as tensor fields. Since diffeomorphisms act on tensor fields (by the push-forward), the group Diff(5) acts on Hs. The quotient space Hs/T)iffo(S) by the subgroup of diffeomorphisms isotopic to the identity is called the Teichmiiller space of S and is denoted by 7^. Clearly, the quotient group Mods = Diff(5)/Diffo(5) acts on Ts and the quotient space Ts/Mods is equal to Hsl DiffC^), which is called the moduli space of hyperbolic structures on S and is denoted by Ms. Hence Ts/Mods = Ms. Sometimes (cf., for example, 5.5) it is more convenient to consider another version of Teichmiiller spaces, namely the one related to so-called punctured surfaces. By a punctured surface R we understand a closed (orientable) surface together with a finite subset of it. The points of this finite subset are csLllcd punctures and we denote by R° the result of deleting all punctures from the surface. Let /? be a punctured surface such that /?° admits a complete Riemannian metric of constant curvature — 1. Again, it is well known that such metrics exist if and only if /?° has negative Euler characteristic. We denote by HR the set of such metrics. This set also can be endowed with a natural topology. This time, the group Diff(/?°) acts on HR, and the quotient space Hs/DiffoiR"^) by the subgroup of diffeomorphisms isotopic to the identity is called the Teichmiiller space of R and is denoted by TR . Naturally, the group Mod/? = Diff(/?°)/ Diffo(/?°) is called the mapping class group of R. It naturally acts on TR and the quotient space TR/Mods is equal to H/?/Diff(/?°), which is called the moduli space of hyperbolic structures on R and is denoted by MR . Clearly, TR/Mod R = MR. Note that there is no real difference between mapping class groups of compact and punctured surfaces. If 5 is a compact surface, then we can collapse each component of dS into a point (one for each boundary component) and get a closed surface. Images of boundary components form a finite subset of this surface, and the resulting surface together with this finite subset is a punctured surface. Let us denote it by R. Clearly, S\dS = R°. This leads to a natural map Diff(5') -^ Diff(/?°). The homomorphism Mod^ -^ Mod/? induced by it is easily seen to be an isomorphism.

5.2. Length functions and Fenchel-Nielsen flows Let Of be a vertex ofC{S), i.e., the isotopy class of a nontrivial circle. For any metric h e Hs the isotopy class a contains exactly one geodesic with respect to the metric h. Let us denote by la(h) the length of this geodesic. Clearly, this length depends only on the image of h in 7^. Hence, we can consider la as a function Ts -^ R>0' We call this function the length function corresponding to a. For a simplex a = {c^i,..., of,^} of C(5') we may consider the map Lcj'.Ts^^ R'"Q given by the formula Lcj (x) = {la\ ( x ) , . . . , la,„ {x)}. Clearly, it comes from a similar map La'.Hs^^ ^>o- ^^^ ^^P ^^ ^^ especially important when a is a simplex of maximal dimension. In this case it turns out to be a principal bundle over R!^^Q having as the structure group the additive group R^.

572

N.V.Ivanov

Let us describe the corresponding action of R'" on Ts. Let h e Hsbesi metric representing a point X eTs, and let C/ be the geodesic circle with respect to h in the isotopy class Of/ (1 ^ / ^ m). If we cut S along this geodesic C/, we get a new surface (with a metric of curvature — 1 and geodesic boundary), which has two new boundary components resulting from Ci. In order to restore S from this new surface, we have to glue back these two boundary components. Clearly, these boundary components should be glued by an isometry, but this does not define the gluing uniquely. Two isometric gluings may differ by a rotation. It seems at first sight that the set of all possible isometric gluings is parameterized by a circle. In fact, since we are interested in the resulting point of Ts (and not only of Ms), i.e., we are interested in the resulting metric up to isotopy (i.e., up to the action of Diffo(5')) and not only up to isometry, the set of all possible gluings is parameterized by the universal cover of this circle, i.e., by a copy of R. More precisely, there are (at least) two natural identifications of this universal cover with R: the one corresponding to the measuring of the amount of rotation by the angle, and the other corresponding to the measuring of the amount of rotation by the length (one can get from the first to the second by multiplication by lai (h)/2n). Another ambiguity comes from the need to orient the circle C/ in order to get the identification, but none of these is of any importance for us now. To summarize, we may cut the original surface and then glue it back by a gluing differing from the one leading back to the original surface by a parameter r G R. This defines a map RxTs -^ Ts, and it is easy to see that this map is an action of the additive group R. This action is usually called a Fenchel-Nielsen flow. Of course, this action depends on the vertex a/. For any h e Hs, the corresponding circles C/, 1 ^ / < m, are disjoint (because a is a simplex), and this easily implies that the actions corresponding to different vertices of cr commute. Hence, they define an action of R^. The fact that this action turns L^ into a principal bundle amounts, essentially, to the following. In the notation of the previous paragraph, if the simplex a is maximal and we cut 5 along all circles C/, 1 ^ / < m, we will get a collection of discs with two holes. A metric of the constant curvature — 1 with geodesic boundary on a disc with two holes is uniquely determined, up to isotopy, by the lengths of its boundary components. Hence, the cut surface is determined by L^ (x). Clearly, if two surfaces (with metrics) lead to the same cut surface, then they differ only by the gluing, i.e., by the action of R^. It follows that the orbits of R'^ are exactly the fibers of the map L^. One needs also to check that the action of R'" on Ts is free, and also the existence of local sections. We refer to [53, Expose 7] for this.

5.3. The topology of TeichmUller spaces As we just saw, the TeichmUller space Ts is the total space of a principal bundle over R^Q with the fiber R^. Since R^Q is contractible, it follows that Ts is homeomorphic to R^Q X R'", and hence to R^'^\ The number m is the maximal number of pairwise disjoint pairwise non-isotopic non-trivial circles on S, hence m = 3g — 3-\-b, where g is the genus and b is the number of boundary components of S. We conclude that Ts is homeomorphic to R6^-6+2/^.

In addition, one can introduce a smooth structure on Ts in a number of ways, all leading to the same smooth structure. This smooth structure can be characterized by the fact that all

Mapping class groups

573

length functions La are smooth. In fact, a finite number of them is sufficient to embed Ts in a Euclidean space R^ as a smooth submanifold. The straightforward approach requires 9g — 9 + 3b length functions (cf. [53, Exposes 7 and 11]). But for a closed S it is sufficient to use only dim Ts-\-l length functions. And if we consider another version of the Teichmiiller space 7^, in which the lengths of the boundary components are not fixed as in 5.1, then for non-closed S it is sufficient to use only dim Ts length functions. See Schmutz [209] and the references there for these results. No matter what definition of the smooth structure is used, it is easy to see that the principal bundle from 5.2 is smooth. It follows that Ts is not only homeomorphic, but also diffeomorphic to a Euclidean space (no results about topological manifolds need to be used!). Also, the Fenchel-Nielsen flows are smooth. Moreover, we can replace smooth by real analytic everywhere. For closed S the Teichmiiller space Ts even has a canonical complex analytic structure, but it plays no role in the sort of questions we are interested in (applications to the mapping class groups). The discussion in 5.2 and in this section applies equally well to Teichmiiller spaces of punctured surfaces. In particular, Teichmiiller space TR of a punctured surface R is diffeomorphic to R6^-6+2/?^ where g is the genus of R and b is the number of punctures, unless /? is a torus without punctures, in which case TR is diffeomorphic to R^. 5.4. Comers and the mapping class groups As we noticed in 5.1, Mod^ acts on Ts. This action is not free, because Mod^ has torsion and Ts is homeomorphic to a Euclidean space (any element of finite order has a fixed point in Ts, say by the Smith theory - see, for example, [53, Expose 11, end of §IV]). But it is as close to being free as possible: any torsion free subgroup F of Mod^ acts freely on Ts (the argument goes as follows: if the isotopy class of a diffeomorphism F fixes a point of Ts represented by a metric h e Hs, then F is isotopic to an isometry F^ of the metric h; it is well known that the group of isometrics of a metric of constant curvature — 1 on 5 is finite). It is well known that Mod^ contains torsion free subgroups, even of finite index. In fact, the kernel of the natural homomorphism Mod5 -^ Aut(H\ (S, Z/mZ)) defined by the action of diffeomorphisms on homology is such a subgroup for any m ^ 3. This result is a combination of a theorem of Serre (cf. Theorem 6.8.A) and a classical result of Nielsen (cf. Theorem 7.1.A) about realization of finite cyclic subgroups of Mod^ by subgroups of Diff(5'); see Corollary 7.I.B. In addition, this action is properly discontinuous. Hence, for torsion free F the quotient space Ts/F is a K(F, l)-space and, of course, a manifold. It turns out that this manifold is never compact, but one can construct a compact manifold from it if F is of finite index in Mod^ either by adding some boundary at infinity or by deleting a neighborhood of infinity. Moreover, one can do this in a fairly canonical way. Before we describe these constructions, we note that the resulting compact manifolds with boundary will not be smooth manifolds in the usual sense, but smooth manifolds with corners. Recall that a smooth manifold with corners is simply a smooth manifold modeled on the products Ry^ x R^ for all n, m (and not only for n = 1 as manifolds with boundary). The simplest type of such manifolds is presented by the subsets of usual smooth manifolds locally described by inequalities of the form x\ > 0 , . . . , jc,, > 0, where ( x i , . . . , x,7,..., Xn^m) is a chart.

574

N.VIvanov

Let us fix some small e >0 and let us assume (only for convenience) that the boundary length / fixed in 5.1 is equal to s. Let Ts{s) be the subset of Ts defined by the inequalities laM ^ £ for all isotopy classes a of non-trivial circles. Clearly, Ts(s) is invariant under the action of Mod^ on Ts. The basic properties of Tsie) are summarized in the following theorem. 5.4. A. Ifeis sufficiently small, then the following holds. (i) The action of Mods on Ts(s) is properly discontinuous. (ii) Any torsion free subgroup F of Mods acts on Ts(£) freely, and its quotient space Ts(e)/r is a smooth manifold with comers. (iii) The quotient space Ts(£)/Mods is compact. (iv) Ts(6) is contractible. (v) The boundary dTsis) is homotopy equivalent to the (geometric realization of the) complex of curves C(S).

THEOREM

PROOF. First, the e should be chosen so small that the following holds: if /i is a metric of constant curvature —1 on 5 and Ci, C2 are two geodesic circles (with respect to /z) on 5 of length ^ 8, then C\, C2 are either equal or disjoint. The existence of such an £ is well known; cf. [1, Chapter II], corollary of Lemma 1 in Section 3.3; it also follows from [29, Corollary 4.1.2]. The property (i) and the first part of the property (ii) immediately follow from the corresponding properties of Ts itself. Property (iii) follows from the so-called Mumford compactness theorem [187]; see [29, Theorem 6.6.5] or [1, Chapter II, theorem in Section 3.4] for a closely related result. The proof of (iv) is the most technical part of the proof of the theorem; we refer to [106, Theorem 3] for a proof. Next, we turn to the property (v). For any vertex a of C(S), let us consider the subset Ts(s, a) = {x e Ts(e) \ la(x) = s] of Ts(e). This subset turns out to be closely related to Ts^(£), where Sa is the result of cutting S along a representative of a, and one can use this together with (iv) for Sa in order to prove that Ts(£, a) is contractible. It is easy to see that the boundary of Tsie) is equal to the union of subsets Ts{s, a). By the choice of e, any collection Tsis, of/) of such subsets has a nonempty intersection if and only if the vertices of, can be represented by disjoint circles. Hence, C(S) is the nerve of a cover of dTsis) by contractible subsets. The property (v) follows. Finally, the second part of the property (ii) follows from the fact that Tsie) itself is a manifold with comers. This follows, in turn, from the fact that in a neighborhood of a point x e dTsie) the subset Tsis) is defined by di finite number of the inequalities of the form lai > s and, moreover, the corresponding a/ form a simplex, and hence can be included as coordinate functions in a chart (by 5.2). A more detailed outline of the proof is contained in [98, Section 4], with complete details provided in [106]. An outline of another proof is contained in [79, Section 3]. D COROLLARY

5.4.B. Mod^ is finitely presented.

Let r be a torsion free subgroup of finite index in Mod^. It follows from (i) and (iv) that Tsis)/r is a KiF, l)-space. Because F is of finite index in Mod^, it follows from (iii) that Tsis)/F is compact. In view of (ii), Tsis)/F is a smooth manifold with comers. The last property implies that Tsis)/F is homotopy equivalent to a finite PROOF.

Mapping class groups

575

CW-complex (moreover, it admits a finite triangulation). Hence, there exists a finite CWcomplex which is a K(r, l)-space. It follows that F is finitely presented. By Lemma 4.3.C, Mods is also finitely presented. D The idea behind Theorem 5.4.A is due to Harvey [86,87]. In [87], Harvey announced a construction of a manifold with comers Xs, containing the Teichmiiller space Ts as its interior and such that the properties (i)-(v) of Theorem 5.4.A hold with Ts(s) replaced by Xs' He was motivated by the analogy (first noticed also by him) between the mapping class groups and arithmetic groups (cf. 9.1) and by the work of Borel and Serre [23]. Adding the comers to Ts in this manner tumed out to be a more delicate matter than excising a part from Ts as in Theorem 5.4.A. A proof of the existence of such an Xs was provided in [105], together with a geometric interpretation of the points in the complement Xs\Ts = dXs.

5.5. Ideal triangulations Let /? be a punctured surface in the sense of 5.1 with a non-empty set of punctures. An ideal triangulation of /? is a collection of (embedded) arcs with disjoint interiors connecting the punctures of R such that if we cut R along all these arcs, we will get a collection of triangles (with sides resulting from these arcs). The word ideal is justified by the standard understanding that the punctures are points at the infinity. In other words, we ideally triangulate /?° rather than R. Note also that two endpoints of an arc are permitted to coincide and a triangle of an ideal triangulation is not determined by the set of its vertices (which are punctures) in general. The ideal triangulations, considered up to isotopy, are exactly the top dimensional simplices of a simplicial complex A{R), which we now define. The vertices of A{R) are the isotopy classes of non-trivial arcs connecting punctures of R. Here an arc is called trivial if two its endpoints coincide and the resulting circle bounds a disc in R which does not contain other punctures. The isotopies are permitted, naturally, only in the class of arcs connecting punctures, so they are fixed on the endpoints. The simplices are defined in a manner similar to the simplices of complexes of curves: a collection of vertices forms a simplex if these vertices can be represented by arcs with disjoint interiors. Like complexes of curves, A{R) is diflag complex (cf. 3.1). As in the case of complexes of curves, this can be established by using complete metrics of constant curvature —1, this time on /?°, and geodesic arcs. Clearly, Mod/? acts on A{R). We also need a subcomplex A^oiR) of A{R). It has the same vertices as A{R). A collection of vertices forms a simplex of Aoo(R) if those vertices can be represented by arcs with disjoint interiors such that the result of cutting R along these arcs has at least one component which is not a disc meeting the set of punctures only in its boundary. In other words, simplices of A(R) which are not simplices of Aoo(R) correspond to the isotopy classes of ideal cell decompositions of R in an obvious sense. 5.5. A. Let A^~^ he the open standard (^ — 1)-dimensionalsimplex, where b is the number ofpunctures of R. Let us endow A^~^ with the trivial action ofModR, and the

THEOREM

576

N. V Ivanov

product TR X A^~^ with the product action. Then TR X A^~^ is ModR-equivariantly homeomorphic to the geometric realization of A(R) with the geometric realization o/Aoo(^) removed. Note that the homeomorphism implied in this theorem is not canonical: different proofs lead to apparently different homeomorphisms. The early history of this theorem is not well documented in the literature. According to [79, Chapter 2], the idea of this triangulation is due to Thurston; his approach was based on hyperbolic geometry. The first published account is contained in [78], where an alternative proof, due to Mumford and based on work of Strebel on quadratic differentials on Riemann surfaces (cf. [218,219]), was outlined. Proofs based on hyperbolic geometry were pubhshed by Penner [197] and by Bowditch and Epstein [24]. Penner's approach seems to be more remote from the original ideas of Thurston in that the Lorentz model of the hyperbolic plane plays a crucial role in it. His ideas were extended, to some degree, to the higher dimensions [50] and to closed surfaces [189], and in [202] Penner applied them to the so-called universal Teichmuller space. Unfortunately, all this falls outside the scope of the present paper. The complexes A(R) and Aoo(R) are, together with complexes of curves, the basic combinatorial objects of the topology of surfaces. To a large extent they are simpler than complexes of curves. Theorem 5.5.A being the main reason, but they are not defined for closed surfaces. Among the main applications of the complexes A(R) and Theorem 5.5.A are the computation of the virtual Euler characteristic of the mapping class group by (independently) Rarer and Zagier [85] and Penner [198,199] (cf. 6.7), computations and estimates of the Weil-Petersson volumes of Teichmuller spaces by Penner [201], Theorem 8.1.A about automorphisms of complexes of curves (cf. 8.4), the proof of Witten's conjecture [233] by Kontsevich [126] and the theorem of Mosher [181,182] to the effect that the mapping class groups are automatic in the sense of [49]. For an exposition of the work of Kontsevich we refer to [141,142], and for an introduction to Mosher's results we refer to his own survey [183]. As an easy and useful corollary of Theorem 5.5.A we point out the following result. COROLLARY 5.5.B. Every simplex of A(R) is a face of a top dimensional simplex. Every codimension 1 simplex of A(R) is a face of I or 2 top dimensional simplices. Any two top dimensional simplices A, A! of A(R) can be connected by a chain of simplices A = A\,..., Am = A' such that any two consecutive simplices Z\/, ^/+i have a common codimension I face.

A much more elementary proof of this corollary was provided by Hatcher [89]; cf. [89, corollary].

6. Cohomological properties This section is devoted to cohomological properties of mapping class groups. It is centered around the computation of the so-called virtual cohomological dimension of mapping class

Mapping class groups

577

groups (see 6.4). As a preparation for this computation, we present in 6.1 a very short and idiosyncratic introduction to the cohomology of groups. This introduction is heavily slanted toward phenomena crucial for understanding of both mapping class groups and arithmetic groups (we will discuss the latter in 9.1) and known as the Bieri-Eckmann duality (cf. [13]). The latter is actually a manifestation of the Poincare-Lefschetz duality, as we try to explain in 6.1. After this introduction, we prove in 6.2-6.4 Harer's theorem [76,78] computing the virtual cohomological dimension of the mapping class groups. In 6.5-6.7 we review, mostly without proofs, some related cohomological properties of mapping class groups. Finally, in 6.8 we review some results about cohomology of the mapping class groups with finite coefficients.

6.1. Cohomology of groups and the Poincare-Lefschetz duality The goal of this section is to present an introduction to cohomology of groups and a fragment of the theory of Bieri and Eckmann [13] (see, especially. Theorem 6.1.H). Let r be a discrete group and let X be a K{r, l)-space with a fixed isomorphism F -^ n\ {X), which will be used to identify these groups. The homology or cohomology of the group r can be defined as the homology or cohomology of the space X. In order to capture the essential properties of F , one has to consider also homology and cohomology with socalled twisted coefficients. Let M be a F-module (in other words, M is a module over the group ring Z[r]). Then the homology //*(r, M) and cohomology H'^iF, M) of F with (twisted) coefficients M can be defined as follows. For a space Y let us denote by C^{Y) its singular complex, or, if F is a CW-complex, its complex of cellular chains. Let X^ ^ X be the universal covering of X. The fundamental group F = 7T\ {X) acts naturally on X^, turning C*(Z^) into a complex of F-modules. By definition, //^(r, M) = H^(X, M) = //*(C*(X~) O r M ) , where <S>r means the tensor product over the group ring Z[F]. If M = Z with the trivial structure of /"-module (i.e., ym = m for ally e F,m e M), then C*(X^) (S^rM = C^(X^) O r Z = C*(X) and we recover the usual homology of F and X. (In fact, if two singular simplices or cells c\, C2 in X^ are projected to the same simplex or cell in X, then c\ = yc2 for some y e F, and hence c\<S^m = yc2<S)m =C2<S> ym = c2^m for any m e M.) Cohomology is defined in a similar way: / / * ( r , M) = //*(X, M) = 7/*(Homr(C*(X'^), M)), where Homr(C*(X^), M) C C*(X^, M) is the group of equivariant M-valued cochains, i.e., the group of cochains / such that f(yc) = yf(c) for any y e F and any chain c. As above, if M = Z with the trivial structure of T-module, we recover the usual cohomology ofTandX.

578

N.VIvanov

I': we choose a lift to X^ of any singular simplex or cell in X, then any equivariant cochain will be defined by its values on these lifts. It follows that Homr(C*(Z'"), M) and C*(X, M) are isomorphic as graded Abelian groups. But the differentials are different (in the first complex, the differential is twisted). Now, let us define the cohomological dimension cdF of F as the supremum (which can be oc) of the numbers n such that H^ (F, M) ^ 0 for some T-module M. THEOREM

6.1 .A. IfX is a closed topological manifold of dimension n, then cdF = n.

PROOF. First, note that H^(X, M) = 0 for m > « by same reasons as for trivial coefficients. Thus we only need to find a T-module M such that H^(X, M) ^^.\f X is orientable, then we can take M = Z with the trivial structure of T-module, otherwise we should take the orientation module Zor- It is isomorphic to Z as an Abelian group, but y e F acts on it by multiplication by —1 if y reverses the orientation of X^, and by 1 otherwise. D

This proof at least partially motivates the need for twisted coefficients in the definition ofcdr. As usual, a short exact sequence of coefficients 0 ^ M' ^ M ^ M'' ^ 0 leads to a long exact sequence of cohomology groups: > H^'iF, M') -^ H^'iF, M) -^ H^'iF, M") -^ H^'^^F, M') -> • • •. It is the key tool in the proof of the following lemma. LEMMA 6.1 .B . If cdF < oo, then cd F is equal to the supremum of numbers n such that H"" ( r , M) 7^ Ofor some free Z[F]-module M. PROOF.

Let n =cdF. Choose M such that H^{F, M) ^ 0 and choose a short exact se-

quence

with F free. Then we have an exact sequence / / " ( r , F) -^ H'^iF, M) -^ H^'^^F, where H''(F,M)

H''iF,F)^0.

/ 0 and H^'^^F^M')

M'),

= 0 because n + 1 > cdT. It follows that

D

LEMMA 6.I.C. / / X is a finite CW-complex, then H^'iF, F) =0 for all free Z[F]modules F if and only ifH^'iF, Z[F]) = 0.

Mapping class groups

579

The "only if" part is trivial. In order to prove the "if" part, suppose that / / " ( r , Z [ r ] ) = 0. Then, for any finitely generated free module F = (Z[r])^, we have

PROOF.

//"(r, F) = H^'iF, (Z[r])^) = H"(r, z[F]r = o'" = o. Now, any free module is equal to the direct limit of its finitely generated free submodules. As we will explain in a moment, the fact that Z is a finite CW-complex implies that the functor H^{F, •) = H^(X, •) commutes with the direct Hmits. Hence, the vanishing of the cohomology with coefficients in finitely generated free modules implies the vanishing of the cohomology with coefficients in an arbitrary free module. This completes the proof, modulo the claim about commuting with the direct Hmits. It is well known that for any ring R (we are interested in the case R = Z[F]) and any projective /^-module Q the functor Q 0/? • commutes with the direct limits. In addition, if P is a finitely generated projective /^-module, then Hom/?(P, M) — P* (g)/? M for any /?-module, where P* = Hom(P, R). It follows that for a finitely generated projective P-module F the functor Hom/?(P, •) commutes with the direct Hmits. Returning to our situation, notice that if X is a finite CW-complex and we consider cellular chains, then Q (X^) are finitely generated free (and hence projective) Z[r]-modules. Thus the result of the previous paragraph implies that the functor Homr(CA:(X^), •) commutes with the direct limits. It remains to note that taking the (co)homology groups of a complex always commutes with the direct limits. D COROLLARY 6.1 . D . IfXis a finite CW-complex, then cd F is equal to the maximum of numbers n such that H'^iF, Z[F]) ^ 0.

Since X is a finite CW-complex, dimX < oo. If we use cellular chains, then CkiX^) =OfoYk> dimX, and hence / / ^ ( r , M) = //"(X, M) = 0 for k > dimX. It follows that cd r < oo. It remains to apply Lemmas 6.1 .B and 6.1 .C. D PROOF.

The last corollary shows the importance of the groups //" (F, Z[F]). Our next goal is to provide a geometric interpretation of these cohomology groups. 6.1 . E . For any F-module M the group Homr (M, Z[F]) is naturally isomorphic to the group Homc(M, Z) of all homomorphisms ofAbelian groups f \M ^^Z such that for any m e M the image f(ym) is equal to 0 for almost all y e F (i.e., for all y with the exception of finitely many of them). LEMMA

PROOF.

Any homomorphism F :M -> Z[F] of T-modules can be written in the form F(m)=

^fy(m)y yer

for some homomorphisms fy'.M^^Z. Clearly, for any m the image fy (m) is equal to 0 for almost all y G T (by the definition of Z[F]). Since F is a homomorphism of F-modules, we have F{^m) = pF(m) for any m e M, fi e F.li follows that E/K(iSm)y = ^ / ^ ( m ) ^ y , yer yer

580

N.VIvanov

and hence f^yiPm) = fy(m) for all p,y e F and m e M. In particular, fy(m) = f\ {y~^m) for all y G T, m G M, where 1 G T is the unit element of F. It follows that F is determined by f\. Moreover, because for any m the image f\iym) = fyy-\iym) = fy-\ (m) is equal to 0 for almost all y, the homomorphism f\ belongs to Homc(M, Z). Conversely, for a homomorphism /i : M -> Z belonging to Homc(M, Z), we can define a homomorphism F : M ^^ Z [ r ] of T-modules by the formula F(m)=

^f\{y~^m)y. yer

Thus F \-^ f\ establishes a natural bijection Homr(M, Z [ r ] ) -^ Hom^CM, Z).

D

LEMMA 6.I.F. IfX is a finite CW-complex, then H*(X,Z[F]) is naturally isomorphic to the cohomology group with compact support H*(X^) ofX^. PROOF.

By Lemma 6.1.E, / / * ( r , Z [ n ) = / / * ( X , Z [ n ) = /^*(Homr(C*(X^),Z[r])) = //*(Homc(C*(X^),Z)).

If we use the cellular chains, then Homc(C*(X"^), Z) consists of cochains which are equal to 0 on almost all cells in the preimage of any given cell in X. Because X has only a finite number of cells, these are exactly the cochains equal to 0 on almost all cells in X^, i.e., the cochains with compact support. It follows that the last cohomology group is nothing other than the cohomology groups with compact support of Z ^ . D COROLLARY 6.I.G. If X is a finite CW-complex, then cdF is equal to the maximum number n such that H^(X^, Z) ^ 0.

Now, it is the time to make a crucial step and apply the Poincare-Lefschetz duality. THEOREM 6.I.H. Suppose that X is a compact topological manifold of dimension d with boundary (may be, empty) and, simultaneously, is a finite CW-complex. Let m be the minimum number such that Hmi^X^) / 0, where ^*(-), cis usual, denotes the reduced homology. Then cdF = d — m — I. (If there is no such m, then cdF = 0.)

By the Poincare-Lefschetz duality H^(X^) = Hd-n(X'^, dX^). Because X is a K(F, l)-space, X^ is contractible, and hence Hd-n(X^, dX^) = Hd-n-\(^X^)- It remains to apply Corollary 6.1.G (note that m = d — n — \ii and only ifn = ^ — m — 1). D PROOF.

The assumptions of this theorem are satisfied, for example, when X is a smooth manifold with comers (as in the applications we have in mind) and, hence, can be triangulated.

Mapping class groups

581

6.2. Classifying spaces and universal bundles Here we discuss some auxiliary results about classifying spaces and universal bundles which will be needed in 6.3 and 6.4. Let G be a topological group acting on a topological space X. Let EG -> BG be the Milnor universal principal G-bundle. In particular, EG -^ BG is a locally trivial bundle and its total space EG is contractible. We will denote by EX -> BG the associated bundle with the fiber X. Recall that EX = EG XG X. A choice of a base point x e X defines a map G ^^ X given by the formula g \-^ gx and a map EG -^ EX given by the formula y\-^ y XGX. LEMMA

6.2.A. If G -^ X is a Serre fibration, then the map EG -> EX is also a Serre

fibration. PROOF. Locally, over any sufficiently small open set U C BG, the bundle EG -^ BG is isomorphic to the trivial bundle L^ x G ^- L^, the associated bundle is isomorphic to U x X - > f / a n d t h e m a p £ G ^ £ Z i s t h e m a p [ / x G ^ [ / x X induced by G ^ X. If G ^ X is a Serre fibration, then, obviously, L ^ x G - > L ^ x X i s a Serre fibration also. It follows that EG -^ EX is a Serre fibration locally (over EX). By a well known theorem (see, for example, [25, Theorem VII.6.11]), this impUes that EG -> EX is a Serre fibration. D LEMMA 6.2.B. Let H be the stabilizer of the base point x in G. Suppose that the above map G -> X is a Serre fibration. Then the part

m (X) -^

TV I (EX) -^

TTi (BG)

of the homotopy sequence of the bundle EX -^ BG can be canonically identified with the part 7Ti(X) ^

7to(H) ^

7to(G)

of the homotopy sequence of the fibration G ^

X.

PROOF. Let TTI (BG) -^ 7TQ(G) be the boundary map from the homotopy sequence of the bundle EG -> BG. Since EG is contractible, it is an isomorphism. By Lemma 6.2.A, the map EG -^ EX is a Serre fibration; obviously, its fiber is H. Let 7t\(EX) -> no(H) be the boundary map from the homotopy sequence of this fibration. Again, since EG is contractible, it is an isomorphism. Direct check shows that the following diagram

TTi (X) —^7ri(EX) —^TTi (BG) 7to(X)

^7to(H)

^7ro(G)

is commutative, and hence provides the required identification.

D

582

N.VIvanov

If G acts on another topological space X' and a G-equivariant map X -^ X' i^ given, then there is a natural map EX -^ EX' commuting with the projections EX -^ BG, EX' -> BG. If X ^^ X' is a. (weak) homotopy equivalence, then EX -^ EX' is also a (weak) homotopy equivalence, and we can identify the homotopy sequences of fibrations EX -^ BG and EX' -^ BG.

6.3. Mess subgroups In this section we discuss some subgroups of the mapping class groups discovered by Mess [163]. These subgroups will be our main tool for estimating the virtual cohomological dimension of the mapping class groups from below (cf. the proof of Theorem 6.4.A). Let Sg be a closed orientable surface of genus g, and let Mod^ = Mod5'^,. Let 5' be an orientable surface of genus g with 1 boundary component, and let A1^ = A^^i. This section is devoted to some subgroups Bg, B^ of Mod^, Aig respectively, where g ^ 2. They were introduced by Mess [163], and we call them Mess subgroups. Although the construction of these involves some choices, it is essentially canonical. The construction of the Mess subgroups has a recursive character. We start with any subgroup B2 of Mod2 generated by Dehn twists about any three pairwise disjoint pairwise nonisotopic circles on ^2. Thus B2 is isomorphic to Z^. Suppose that Bg, g ^ 2, is already defined. We may assume that Sg is obtained from SJ, by gluing a disc D^ to the boundary of SJ,. The extension of diffeomorphisms of SJ,fixedon dS^^ by the identity across the disc D^ defines a homomorphism A4^ -^ Modg as in 2.8. Let B^^ be the preimage of Bg under this homomorphism. In order to define Bgj^\, consider some embedding SJ, —> Sg^\ and identify 5'] with its image. The extension of diffeomorphisms of 5* fixed on dS^ by the identity across the complement of 5] in Sg-^\ defines a homomorphism /: Mg -^ Mod^+i. Note that the closure of the complement of ^j in Sg-\-\ is a torus with one hole. Let us choose some nontrivial circle in this torus with one hole and consider the Dehn twist t e Mod^+i about this circle. Let T be the infinite cyclic group generated by t. We define Bg^\ as the group generated by iiBh and T. This completes the construction of the Mess subgroups. A couple of remarks are in order. First, recall (see 2.6) that the restriction of diffeomorphisms of Sg on D^ defines a fibration Diff(5'g) -^ Emb(D^, Sg). As explained in 2.6, its fiber can be identified with Diff(5'^ fix 9). Since Diff(Sg) is simply connected (by Theorem 2.7.G) and Emb(D^, Sg) is connected, the homotopy sequence of this fibration ends with 1 -^

TTi (Emb(D^ Sg)) -^

7ro(Diff(5'] fix 9)) -^

1 -^

TTi (Emb(Z)^ Sg)) — > M l —> Modg —> 1,

i.e., with

;ro(Diff(5^)) —> 1,

Mapping class groups

583

where the homomorphism A^^ -^ Modg is exactly the one used above. Hence, we have a short exact sequence 1 -^

TTi (Emb(D^ Sg)) —> B I ^ Bg —> \.

Since Emb(D^, Sg) is weakly homotopy equivalent to the unit tangent bundle UT(Sg) (by Theorem 2.6.C), we can also write this short exact sequence as

1 -^ ir, {UT(Sg))

-^Bl^Bg^X.

Note also that the homomorphism / '-Mi -^ Mod^+i is injective by Theorem 2.7.1 and that the twist t obviously commutes with the image of/. It follows that Bg^\ is isomorphic to B^ X Z. Now we come to the main property of the Mess subgroups (cf. [163, Proposition 1]). 6.3 .A. There exists a closed topological manifold of dimension 4g — 5 which is a K(Bg, I)-space. Similarly, there exists a closed topological manifold of dimension 4g — 2 which is a K(B^, I)-space. These topological manifolds can be chosen to be simultaneously CW-complexes. THEOREM

The idea is to start with the 3-dimensional torus, which can be taken as a K(B2, l)-space and to construct a K(B^, l)-space as the total space of a bundle with the fiber UT(Sg) over an already constructed K{Bg, l)-space, and then get a K{Bg^\, 1)space by multiplying this total space by the circle S^. We would like this bundle to realize the sequence 7T\ (UT{Sg)) -> 5 ] -> Bg as the 7t\ -part of its homotopy sequence. Note that UT{Sg) is a K{7i, l)-space, because it is the total space of a circle bundle over Sg, which is a K(Tt, l)-space. Suppose that a K(Bg, l)-space Kg is already constructed. Let G = Diff(5^0, X = Emb(D^, Sg) with the obvious action of DiffC^^,), and let X' be the unit tangent bundle UT(Sg) with the following action of Diff(5'^,)* a diffeomorphism acts on a unit vector by its differential and the subsequent normalization. Let X ^^ X' be the map u : Emb(D^, Sg) -> UT(Sg) from 2.6. It is obviously G-equivariant. By Theorem 2.6.C, it is a weak homotopy equivalence. Hence the induced map EX -^ EX' is also a weak homotopy equivalence. It follows that we can naturally identify the homotopy sequences of bundles EX -^ BG and EX' -^ BG. Now, since components of Diff(5'g) are weakly contractible by Theorem 2.7.G, the classifying space BG = B Diff{Sg) is a Ar(Mod^, l)-space. Hence, the inclusion Bg -^ Modg leads to a map Kg -^ BG from our K{Bg, l)-space Kg to BG = B Diff(Sg). Consider the bundle K^ -^ Kg with the fiber X' = UT(Sg) induced from the bundle EX' -^ BG by this map. Since both UT(Sg) and Kg are K(7T, 1)-spaces, the homotopy sequence of the induced bundle shows that K^ is also a K(n, l)-space. In addition, this homotopy sequence ends with the short exact sequence PROOF.

1 -^

ni{UT(Sg)) -^

7ti{Kl) -^

7ti(Kg) ^

I.

584

N.VIvanov

This short exact sequence naturally maps to the end 1 -^

7l\ {UTiSg)) —> TTl {EX') —> TTl (BG) —> 1

(recall X' = UT(Sg)) of the homotopy sequence of the bundle EX' -^ BG. Moreover, n\(Kh is obviously isomorphic to the preimage of 7t\(Kg) = Bg under the map TTl (EX') -^ 71 \ (BG). By the previous paragraph, this short exact sequence is naturally isomorphic to the end 1 -^

7Tx(X) -^

ni(EX) - ^ 7ti(BG) -^

1

of the homotopy sequence of the bundle EX -^ BG. Now, consider the fibration G = Diff(Sg) -> X = Emb(D^ Sg). Clearly, H = Diff(5] fix 3) is the stabiHzer of the embedding D^ -> Sg and, hence, is the fiber of this fibration. By Lemma 6.2.B, the last short exact sequence is naturally isomorphic to the end 1 -^

7ti(X) -^

7To(H) -^

no(G) —> 1

of the homotopy sequence of the fibration G ^- X. But this short exact sequence is nothing other than our sequence 1 —> 7ri(Emb(D^ Sg)) —^ Ml ^

Modg -^

1.

It follows that 7T\ (Kh is isomorphic to the preimage of Bg under the map /: Ml -> Mod^, i.e., to B^. Hence, we can take K^ as our K(B^^, l)-space, and then put Kg-^\ = K^^ x S\ This completes our construction of K(Bg, 1)- and K(B^^, l)-spaces. Since they are constructed by consecutively taking the total spaces of locally trivial bundles with 3-manifold fibers and multiplying by S^ (and starting with the 3-dimensional torus), these spaces are, obviously, topological manifolds. With some additional care, one can turn these spaces into CW-complexes (or even smooth manifolds, and then triangulate them). This proves all statements of the theorem with the exception of the correctness of the dimension values. Clearly, dimA^j = 6miKg + 3, dimA'^+i = d i m ^ j + 1 and dim ^2 = 3. Therefore, dim Kg=Ag — 5 and dim A^j =4g — 2. This completes the proof. D COROLLARY PROOF.

6.3.B. cdBg =4g-5,

cdB] = 4g - 2.

Combine Theorem 6.3.A with Theorem 6.1 .A.

D

6.4. Virtual cohomological dimension One of the main properties of the cohomological dimension is the inequality cdF' ^cdf, which holds for T^ C T. We refer to [26] for a proof and only note that it uses in a crucial way the fact that all twisted coefficients are allowed; cf. [26, Chapter VIII, Proposition 2.4].

Mapping class groups

585

It is well known that cd A = oo for a finite cyclic group A; it follows that any group with torsion has infinite cohomological dimension. In order to get an interesting invariant for groups with torsion, we modify the definition in the way suggested by Serre [213]. If r is virtually torsion free, i.e., contains a subgroup of finite index which is torsion free, then, by a fundamental theorem of Serre [213] (cf. also [26, Chapter VIII, Theorem 3.1] or [11, Section 3]) all torsion free subgroups of finite index of F have the same cohomological dimension. The common value of the cohomology dimension of such subgroups is called the virtual cohomological dimension of F and is denoted by wcdF. Since Mods always has torsion, cdMods = oo and the interesting invariant is vcdMod^. The results of 6.1 and 6.3 allow us to compute it. We start with closed surfaces. 6.4. A. If S is a closed surface of genus g^2, then vcdMod^ =4g — 5.IfS is a sphere or a torus, then vcdMod^ = 0 or I respectively.

THEOREM

PROOF. If 5 is a sphere, then Mods = 1 and the result is trivial. If 5 is a torus, then Mods is isomorphic to SL2(Z). The last group, as is well known, contains a finitely generated free group as a subgroup of finite index. It is easy to see that the cohomological dimension of a free group is 1. Hence, vcdMods = 1 if 5 is a torus. In the rest of the proof we assume that 5 is a closed surface of genus g ^ 2. Consider a subgroup F of finite index in Mods acting freely on the Teichmiiller space Ts (such subgroups do exist by 5.4). It is sufficient to show that cd F = 4g — 5 for any such F (we do not need to know in advance that such a group F is torsion free, since cd F < oo implies this). Consider the intersection F^ = F HBg of F with the Mess subgroup Bg. It is obviously of finite index in Bg. Hence, we can take as a K(F\ l)-space a finite sheeted covering space of the K(Bg, l)-space provided by Theorem 6.3.A. Clearly, this K(F\ l)-space is a closed manifold of dimension 4^ — 5. By Theorem 6.1.A, cdT^ = 4g — 5. It follows that cdr^4g-5. In order to prove the opposite inequality, consider the manifold with comers Ts(e) from 5.4 and put X = Ts(s)/F. Since the action of F on Ts is free, X is also a manifold with comers, and since Ts(s) is contractible (cf. 5.4), we can take Ts(s) as the universal cover X^ of X. By 5.4, the boundary 9 7s (e) is homotopy equivalent to (the geometric realization of) the complex of curves C(S). Hence, Theorem 3.2.C impHes that dX^ = dTsis) is 2g — 3-connected. Now we apply Theorem 6.I.H. Since dX^ is 2g — 3-connected, the number m from this theorem is ^ 2g — 2 (such an m exist, because we already saw that cd r / 0). Finally, d = dim X^ = dim Ts(e) = dim Ts = 6g — 6, and hence Theorem 6.1 .H impHes that cd T ^ (6g — 6) — (2g — 2) — 1 = 4g — 5. This completes the proof. D THEOREM 6.4.B. If S is a surface of genus g^2 vcdMods =4g -4-\-b.

with b^ I boundary components, then

PROOF. Let Sg^b be a surface of genus g with b boundary components, and let Modg^b = Mods^^. In order to compute vcdMod^,/,, we will constmct a subgroup Fg^b of finite index in Modg^b and a K(Fgb, l)-space. Since the dimension of this space will be < oo, and hence cd Fgb will be < oo, the subgroup Fg^b will be automatically torsion free, and

586

N.VIvanov

therefore we will have vcdMod^,/, = cdFg^b- The cohomological dimension cd Fg^b will be computed with the help of Theorem 6.1.H using our KiPg^b, l)-spaces. The construction of subgroups Fg^b has a recursive character. We start with any subgroup Fg^o of finite index in Mod^,o acting freely on the Teichmiiller space TS^Q. Let Lg^o = Ts^f^(6)/Fg^o^ where Ts^^ie) is the manifold with comers from 5.4; L^,o will be our K(Fg^o, l)-space. Suppose that a subgroup Fg^b of finite index in Mod^,^ and a K(Fgb, 1)-space L^,/, are already constructed. Let us fix a point x in the interior of Sg^b- Let * = {x}; it is a 0-dimensional submanifold of Sg^b- Clearly, we can identify Emb^(*, Sg^b) with the interior iniSg^b of Sg^b (recall that the embeddings in Emb^(*, Sg^b) can be extended to diffeomorphisms of Sg^b\ see 2.6). Now, consider the group Modg^(b,\) = 7ro(Diff(5^,/, fix *)). In view of the last paragraph of 2.8, the group Modg^(b,\) is isomorphic to the subgroup of the group Mod^/,+i consisting of the isotopy classes of diffeomorphisms Sg^b-\-\ -^ ^g,b-\-\ preserving (setwise) one fixed boundary component. In particular, Mod^ (/, i) is isomorphic to a subgroup of finite index (actually, of index Z? + 1) of Mod^,/,+ i. Let G = Diff(Sg^b) and let X = iniSg^b with the obvious action of G. The map G —> Z given by g \-^ gx is nothing but the evaluation of diffeomorphisms at x; it can be identified with the map Diff(5^,^) -^ Emb'^(*, Sg^b) from 2.6. In particular. Theorem 2.6.A implies that G —> X is a Serre fibration. The stabilizer H of the point x e X (i.e., the fiber of G -> X) is equal to Diff(5^,^ fix *). The homotopy sequence of this fibration ends with 1 - ^ 7Ti(mSg,b) —^ 7ro(Diff(5^,^fix*)) — ^ 7to{Diff(Sg,b)) —> 1, i.e., with 1 —> 7t\(lniSg^b) —> Modg^(^b,\) — ^ Modg,^ —> 1. By Lemma 6.2.B, this short exact sequence can be naturally identified with 1 — ^ 7ti(X) - ^ 7ti(EX) —^ jTiiBG) —^ 1 (where X = mtSg^b)Next, we would like to replace the open surface int Sg^b^y the compact surface Sg^b- Let X' be the surface Sg^b with the natural action of G = Diff(5g,/?). The inclusion X ^^ X^ is, obviously, G-equivariant and is a homotopy equivalence. In view of the remarks at the end of 6.2, we can identify the last exact sequence with 1 — ^ TTl (X') - ^ m (EX') —^ TTl (BG) - ^ 1 (where X' = Sg^b)- We can write this short exact sequence more explicitly as 1 - ^ JtiiSg^b) - ^ 7ti{ESg,b) - ^ 7Ti{BDiff(Sg,b))

—> 1.

Now, as in the proof of Theorem 6.3.A, we use the fact that BDiff(Sg^b) is a K(Modgb, l)-space, and hence the inclusion Fg^b -^ Mod^,/, leads to a map Lg^b -^

Mapping class groups

587

Diff(5^,/,) fromoir. y:^ '"\ /,^ l)-space Lg^t to BDiff(Sg^b)' Consider the bundle Lg^b+\ -> Lg^b wim me nuc. . ced from the bundle EX' = ESg^b -^ BG = B Diff(Sg^b) by this map. Since both Sg^b and Lg^b are A'CTT, l)-spaces, the homotopy sequence of this bundle shows that Lg^b+\ is also a K(Tt, l)-space. In addition, this homotopy sequence ends with the short exact sequence 1 -^ 7t\(Sg^b) -^ n\(Lg^b+i) -^ 7t\(Lg^b) -> 1. The latter naturally maps to the last short exact sequence of the previous paragraph, and a trivial diagram chase shows that n\ (Lg^b+\) is isomorphic to the preimage of Fg^b under the map n\{ESg^b) -^ T^\{BT>ifi{Sg^b))' In view of the above, this preimage is naturally isomorphic to the preimage of Fg^b under the map Mod^ (^, i) -^ Modg^b- We define Fg^b-\-\ to be equal to this preimage (obviously, it is of finite index in Modg^b-\-\) and take Lg^b-{-\ as our K(Fg^b+\, l)-space. This completes the construction of our subgroups Fg^b and K(Fg^b, l)-spaces Lg^bSince the latter are constructed by consecutively taking the total spaces of locally trivial bundles with surface (in general, with boundary) fibers starting with a manifold with comers, the spaces Lg^b are topological manifolds with boundary and we also can turn them into CW-complexes (compare the proof of Theorem 6.3. A). Obviously, dim Lg^b-\-\ = dim Lg^b + 2 for all b. In order to apply Theorem 6.1.H, we need to analyze the universal covers L J^ of manifolds Lg^b and their boundaries ^L^b' If the universal cover LJ^ is already constructed, we can construct ^J/^+i in two steps. First, take the bundle Mg^b ~^ ^Zb ^^^ ^^^^ ^K^b induced from the bundle Lg^b+\ -^ ^^,/? by the map L^^^^^ Lg^b- Since the base L^^ of this bundle is contractible, this bundle is actually trivial, and hence Mg^b is homeomorphic to Sg^b X LJ^. As the second step, we take the universal cover of Mg^b- This leads to 5^^ X L'^i^, where S^^ is the universal cover of the surface Sg^b- Thus, ^7/,+! is homeomorphic to 5J^ ^ ^7b' ^^^ hence the boundary 3LJ^_^| is homeomorphic to

Now, there are two different cases to consider: b = 0 and b^ I. In the first case, S'^Q has no boundary and 9L^ j is homeomorphic to S^Q xdL^Q. Since 5^Q is contractible, it follows that dL^ ^ is homotopy equivalent to 9L JQ. In the second case, 95J^ is non-empty and actually consists of a countable (infinite) number of components homeomorphic to the real line. Since 5^^ is contractible, it follows that the pair (5"^^, 95?^) is homotopy equivalent to the pair (CZ, Z), where Z is a discrete space consisting of a countable number of points and CZ is the cone of Z. The last pair, in turn, is homotopy equivalent to the pair (R, Z). It follows that SL^j^^^ is homotopy equivalent to

Since the universal covers L J^ are contractible, the last space is homotopy equivalent to a bouquet of an infinite number of suspensions EdL^ ^.

588

N.VIvanov

Now, let d(b) = dimLg^h and let m(b) be the minimum number m such that Hfn(dL^fy) ^ 0 (we assume that the genus g is fixed). As we saw above, d(b + 1) = d(b) -\-2 for all b. The results of the two previous paragraphs imply that m(l) = m(0) and m{b -\- \) = m{b) -\- \ fox b ^ 1. Combining these remarks with Theorem 6.1 .H, we see that cdFgx = c d 7^^,0 + 2 andcd/]^,/,+ i = c d r ^ , ^ + 1 forZ? ^ 1. Since cd/]^,o = vcdMod^o = 4g — 5 by Theorem 6.4.A, the theorem follows. D THEOREM 6.4.C. If S is a sphere with b holes, then vcdMod^ = 0 for b ^ 3 and vcdMod^ = b — 3 for b ^ 3. If S is a torus with b holes, then vcdMod^ = 1 for b = 0 and vcdMod^ = b for b^\. PROOF. The proof is similar to the proof of the previous theorem. Since for a sphere S with ^ 2 holes or a torus S without holes the components of DiffC^S) are not contractible (and hence B Diff(5) is not a ^(Mod^, l)-space), the inductive argument should start with a sphere with 3 holes and a torus with 1 hole. We leave the details to the reader. D

The first nontrivial estimate vcd Mod^ ^ 6g — 9 of vcd Mod^ for a closed surface of genus g was proved in [95,96] (the trivial estimate is vcdMod^ ^ dmTs =6g — 6).\i was deduced from the simply connectivity of C(5), which, in turn, was deduced from the simply connectivity of the Hatcher-Thurston complex [91]. Theorems 6.4.A-6.4.C are due to Harer [76,78]. Another proof was provided by the author [98]. The paper [98] contains also a computation of the virtual cohomological dimension of the mapping class groups of nonorientable surfaces. Both papers [78] and [98] actually prove a stronger result than Theorems 6.4.A-6.4.C. Namely, they prove that Mod^ contains a subgroup of finite index which is a group with duality of some dimension in the sense of Bieri-Eckmann [13] and compute this dimension. The dimension of a group with duality is built into its definition (like the dimension of a smooth manifold or a Poincare space); it is always < oo and turns out to be equal to the cohomological dimension. The last property allows the use of this result for the computation of vcdMod^. We refer to [98, Section 6] for an expository account of this stronger result, and to [26, Chapter VIII], or [11] for more details about groups with duality. The proof of the lower estimate for the cohomological dimension in our proof of Theorem 6.4.A follows the ideas of Mess [163], while the proof of the upper estimate (which turned out to be equal to the lower one) follows the standard approach of [95,96,76,78,98]. The above proof of Theorem 6.4.B seems to be new. Both of these proofs avoid the full power of the theory of groups with duality (a fragment of this theory was included in 6.1) and also do not use the complete description of the homotopy type of C{S) (they use only Theorem 3.2.C, and only for closed surfaces).

6.5. Homology stability In its simplest form, the homology stability for the mapping class groups asserts that the homology group //^(Mod^^) (with trivial coefficients Z) does not depend on S provided S is closed and the genus ^ of 5 is sufficiently large compared to n. The implied restriction on g is usually called the domain of stability. The first result of this sort, with the domain of

Mapping class groups

589

stability g > 3n + 1 , was proved by Harer [76,77]. Such results are similar to, and motivated by, the classical homology stability theorems in the algebraic ^-theory. The simplest of those asserts that the homology group //„(SL^(Z)) does not depend on g provided g is sufficiently large compared to n. The more advanced versions of the homology stability theorems in algebraic A'-theory cover some other natural series of groups, like the special linear groups over rings or the symplectic groups, and usually provide an explicit domain of stability (the role of g is played by some natural parameter). A common feature of the proofs of the homology stability theorems in algebraic ^-theory is the need to include into the picture a wider class of groups than the original infinite series: this is required by the way the induction is arranged in the proofs. The situation for the mapping class groups is similar: one needs to consider the groups Ais for non-closed surfaces S, in addition to the groups Mod^ for closed S. The groups Ais have a natural advantage over our usual version Mod^, namely the existence of a natural map MR ^^ Ms when /? is a subsurface of S. This map is given by the extension of diffeomorphisms of R fixed on dR by the identity to diffeomorphisms of S. In fact, we can completely switch to the groups Ms in this context, because Ms = Mods for closed S (and the homology stability does not hold for the groups Mod^ for non-closed surfaces S; for the second homology group this follows from Theorem 6.6.C). THEOREM 6.5. A. Let R be a connected subsurface of a connected surface S. Let gR be the genus of R. The map Hm(MR) -^ Hf^ {Ms) induced by the natural map MR -^ Ms is an epimorphism if

gR^2n-hl and is an isomorphism if gR^2n-\-

2.

If in addition, S is not closed, then the map HtniMR) -^ Hm(Ms) is an epimorphism if gR >2n and is an isomorphism if gR^2n-\-L The proof of Theorem 6.5. A is fairly technical. In the outline, it follows the Maazen-van der Kallen [119] approach to the homology stability theorems in algebraic A'-theory. On the geometric side, one of the basic tools is the action of the groups A^5 on some versions of the complexes of curves C(S). The higher connectivity of these versions plays a crucial role. On the algebraic side, the central role is played by some spectral sequences associated with an action of a group on a simplicial complex and a carefully arranged induction on the genus and the number of boundary components of S. For an introduction to the ideas of the proof of Theorem 6.5.A, we refer to [98, Section 7] and to [109, Section 1]. The

590

N.VIvanov

details omitted in [98] are provided in [104]. The case of closed S requires an additional argument and is dealt with in [109]. 6.5.B. If S is a closed surface of genus ^ 2n + 2, then up to isomorphism the homology group Hn (Mod^) does not depend on S. COROLLARY

First, recall that Mod^ = A^^ for closed S. If 5, S' are two closed surfaces of different genera, there is no natural homomorphism between the groups Ms, Ms'- In order to compare the homology of A^5, A^^s we can choose two diffeomorphic subsurfaces R, R' of 5, S' respectively and apply Theorem 6.5.A to the maps MR -^ Ms, MR' -> Ms'- If we choose subsurfaces R, R' such that their genus is equal to the lesser of the genera of 5, S', the corollary will follow immediately. D PROOF.

The domain of stability of Theorem 6.5. A and Corollary 6.5.B is substantially better than the Harer's original domain of stability g^3n-\-l from [77]. For non-closed surfaces, the domain of stability g ^ 2n -\- I of Theorem 6.5.A is exactly the same as the best known domain of stability for //„(SL^(Z)). On the other hand, if we consider homology with rational (as opposed to the integer) coefficients, then the domain of stability can be further improved. (A similar phenomenon is well known in the algebraic ^-theory.) 6.5.C. Let R be a connected subsurface of a connected surface S with nonempty boundary. Let gR be the genus of R. The map Hfj(MR, Q) -> Hn{Ms, Q) induced by the natural map MR ^^ Ms is an isomorphism if THEOREM

gR^

3n y*

For odd n, this map is surjective if

This theorem is due to Harer [84], who also proved that the domain of stability of this theorem cannot be improved. The question about the exact domain of stability for integral coefficients remains open. Finally, we mention two other stability theorems. In [80], Harer proved a homology stability theorem for subgroups of the mapping class groups consisting of the isotopy classes of diffeomorphisms preserving some spin structure on the surface (these subgroups obviously have finite index in the corresponding mapping class groups). The main result of [109] is a homology stability theorem for so-called twisted coefficients, i.e., for coefficients in a non-trivial module. Actually, more important than the fact that the coefficients are twisted is that the natural examples of twisted coefficients substantially depend on the surface in question, i.e., are not constant. As the simplest (and typical) example one may consider Hn(Ms, H\{S)), where Ms acts on H\(S) in the natural way. In this special case, we have the following result.

Mapping class groups

591

6.5.D. Let R be a connected subsurface of a connected surface S. Let gR be the genus of R. Suppose that both R and S have exactly 1 boundary component. Then the map Hn(A4R, H\ (R)) -^ Hn{Ms, H\ (S)) induced by the natural maps MR ^^ Ms, H\ (R) -^ H\ (S) is an epimorphism if THEOREM

gR^2n-\-

1,

and is an isomorphism if gR

^2n-\-3.

This is a special case of Corollary 4.9 from [109] (the coefficient system S \-^ H\(S) is of degree 1 in the sense of [109]). Looijenga [143] provided a new proof of a special case of the stability theorem for twisted coefficients of [109]. Namely, he considers only coefficients which factor through the natural representation Ms ^^ Sp^(Z), where g is the genus of 5, and which are rational vector spaces. In this situation he extends the results of [109] to closed surfaces and, more importantly, explicitly computes the stable homology (i.e., the homology in the domain of stability) with such twisted coefficients in terms of the stable homology with trivial coefficients Q. Looijenga's methods are largely algebrogeometric, but he uses also the homology stability theorem for constant coefficients. Surprisingly, homology stability with twisted coefficients does not hold for closed surfaces, even for the first homology group, as the following theorem implies. THEOREM

6.5.E.

If S is a closed surface of genus g ^ 2, then H\(Ms, H\(S)) =

Z/(2g-2)Z. This result is due to Morita [171] (cf. [171, Corollary 5.4]).

6.6. The low-dimensional homology groups We start by computing the first homology group of PMod^. LEMMA

6.6. A. H\ (PMod^) depends only on genus of S.

It is sufficient to show that H\ (PMod^) does not change if we make a hole in S. So, let R be the result of removing from S the interior of a disc embedded in S. Consider the exact sequence PROOF.

TTi {S) -> PMod/? -^ PMods -> 1 from 2.8. If a G TTI {S) is the homotopy class of an embedded loop in 5, then the image of a in PMod/? has the form taf^^ for some circles a,^ m R. Moreover, if a is the homotopy class of an embedded non separating loop, then both circles a, and ^ are non separating; see Lemma 4.1.1. It follows that ta is conjugate to t^, and hence their images in H\ (PMod/?)

592

A^. V Ivanov

are equal. Therefore, the image of a in H\ (PMody?) is 0. Since 7t\{S) is generated by the homotopy classes of embedded non separating loops (if S has non-empty boundary, then some of the standard generators are separating, but they can easily be replaced by two non separating generators), the image of n\ (S) in H\ (PMod/?) is 0. Using the above exact sequence, we see that H\ (PMod/?) is isomorphic to H\ (FMod^). D 6.6.B. Let S be a surface of genus g. The first homology group H\ (PMod^) is equal to 7J/\27J if g = 1, to Z/IOZ if g = 2, and to 0 if g ^ 3.

THEOREM

If 5 is a closed surface of genus 1 (i.e., a torus), then PMod^ is isomorphic to SL2(Z) and the result is well known (to algebraists, at least). If 5* is a closed surface of genus 2, then, probably, the best way to compute H\ (PMod5) is to use the presentation of Mod^ due to Bergau and Mennicke [12] and Birman and Hilden [19]; cf. end of 4.3. Given the presentation, the computation of H\ (PMod5) is routine and can be left to the reader. This proves the theorem for closed surfaces of genus 1 and 2, and then Lemma 6.6.A implies it for all surfaces of genus 1 and 2. In the generic case g > 3 we actually do not need Lemma 6.6.A. In this case, we can embed the sphere with 4 holes 5Q from Lemma 4.1.H in S in such a way that all circles C/, 0 < / < 3, and C/y, 1 ^ / < 7 ^ 3, will be non-separating in S; see Figure 13. For PROOF.

Fig. 13.

Mapping class groups

593

such an embedding all Dehn twists ti, f/y from the lantern relation are conjugate, and hence represent the same element of H\ (PMod^). The lantern relation tot\ /2^3 = ^12^13^23 implies that this element is equal to 0. On the other hand, by Theorem 4.2.C, PMod^ is generated by Dehn twists about non separating circles. All such Dehn twists are conjugate to to, and hence represent 0 in H\ (Mod^). It follows that H\ (PMod^) = 0 . D The path to this nice argument for the genus ^ 3 case, which is due to Harer [75], was fairly long. First, Mumford [186] proved that H\(Mods) is cyclic of order dividing 10 for closed surfaces S of genus > 2. Then Birman [15] computed H\(Mods) for closed surfaces S of genus 1 and 2, and proved that for genus ^ 3 the order of this group is 1 or 2. Finally, Powell [205] proved that this order is actually 1 by means of some complicated computations. 6.6.C. Let S be a surface of genus g with b boundary components. If g ^ 5, thenH2(PMods)=Z^^K THEOREM

This result is due to Harer [75]. One should warn the reader that [75] contains some mistakes, even in the statement of the main theorem (it is asserted in [75] that H2iPMods) = Z 0 Z/(2g — 2)Z for closed surfaces S of genus g). But the general agreement seems to be that Theorem 6.6.C is correct. The main tool in [75] is an action of PMod^ on a modification of the Hatcher-Thurston complex [91] and the spectral sequences associated with this action. Later on, Harer [81] suggested an alternative method to compute the second homology group with rational coefficients, and also extended this new method to the third homology group. The approach of [81] is close in spirit to the proofs of the homology stability theorems 6.5, and uses the actions of the mapping class groups on complexes similar to those used to prove the stability theorems. The spectral sequences associated with these actions relate the homology groups of a mapping class group with the homology groups of the stabilizers of the simplices, which are, in fact, mapping class groups of smaller surfaces. This allows use of an inductive process for the computations. Let us state the main result of [81]. 6.6.D. Let S be a surface of genus g. Hli-Ms, Q) is equal to 0 if g = 2, and to Q if g ^ 3. m{Ms.Q) = 0ifg^6,

THEOREM

Using the same approach, in [82] Harer computed //2(-, Q) for subgroups of the mapping class groups consisting of the isotopy classes of diffeomorphisms preserving some spin structure on the surface, and in [83] he also computed the fourth homology group of PMod^. More precisely, the main result of [83] is the following theorem. THEOREM

HA{MS.

6.6.E. Let S be a surface of genus g with b boundary components. Q ) = Q^ifg^6,b^\

or ifg ^ 10.

The computations involved in the proof of this theorem are extremely complicated. Finally, in [72] Hain and Looijenga report that recently Harer has extended his computations to //5(PMod5, Q). This homology group turns out to be 0 for sufficiently high genus.

594

N.VIvanov

6.7. Virtual Euler characteristic In the spirit of Serre's approach [213], the usual Euler characteristic ch() = ^•>Q rank W (•) is not the right invariant for discrete groups F with torsion, even when it is well defined. One should consider instead the so-called virtual Euler characteristic x (•)• If F contains a subgroup F' of finite index admitting a finite CW-complex as a K{F', 1)space (this implies that F' is torsion free), then, by definition,

x(n = ^y^ch(r'). It turns out that this number does not depend on the choice of F'. In general and in the most interesting cases, x (^) is not an integer, but only a rational number. For the virtual Euler characteristic there is an analogue of the usual formula computing the Euler characteristic of a CW-complex by counting the number of cells of various dimensions. Suppose that F acts on a CW-complex T preserving its CW-structure. Suppose that, in addition, T is contractible, some subgroup F' of finite index acts freely on T (and hence T/F' is a K(F\ l)-space), the number of orbits of cells is finite and the isotropy groups of cells are finite. Then

x(n = J2^-i)dimcr

[ra'-iy

where F^ is the isotropy group of the cell a, [F^ : 1] is its order and a runs over some set of representatives of the orbits of cells. If 7^ acts on T freely, this formula reduces to the usual formula for the Euler characteristic of the K(F, l)-space T/F. For more details about the virtual Euler characteristic we refer to [213] and to [26, Section IX.7]. 6.7.A. If S has one boundary component, then /(Mod^) = —Bjg/lg = ^(1 — 2g), where g is the genus of S, B2g is the 2g-th Bernoulli number and ^ is the Riemann ^ -function. THEOREM

PROOF. Let /? be a once-punctured surface of the same genus as S. By 5.1, Mod^ is isomorphic to Mod/?. Hence, we may consider Mod/? instead of Mod^. We will use the ideal triangulation of the Teichmiiller space TR introduced in 5.5. Let us consider the (first) barycentric subdivision of the complex A(R), and take the union IJ(R) of the barycentric stars of those simplices of A(R) which are not simplices of Aoo{R)This union has a natural structure of a CW-complex with the barycentric stars as cells. One may say that ZiR) is the complex dual to the ideal triangulation of TR from 5.5. It is not hard to see that E(R) is a Mod/?-equivariant deformation retract of TR. In particular, U{R) is contractible. Cells of I!(R) are in one-to-one correspondence with simplices of A(R) which are not simplices of Aoo(R),i.t., with ideal cell decompositions of R. Under this correspondence, the orbits of cells correspond to the topological types of ideal cell decompositions. Since the number of these topological types is obviously finite, the number

Mapping class groups

595

of orbits is also finite. The isotropy group of a cell corresponding to an ideal cell decomposition is, clearly, equal to the group of isotopy classes of diffeomorphisms preserving this ideal cell decomposition, and hence is finite. (On the contrary, the isotropy groups of simplices in Aoo(R) are infinite.) As in the case of Teichmiiller spaces Ts of surfaces with boundary (see 5.4), there are subgroups of finite index in Mod/? acting freely on TR, and hence on I^(R). It follows that we can use the CW-complex E(R) with its natural action of Mod/? in order to compute x (Mod/?) by the above formula. The application of this formula to this situation is not a simple matter at all. The problem is essentially a combinatorial one, since an ideal cell decomposition of R is an essentially combinatorial object, and the corresponding isotropy group is the group of symmetries of this combinatorial object. But there is no simple way to list all ideal cell decompositions (up to homeomorphism, of course) and to find the orders of their groups of symmetries. By some breathtaking combinatorial arguments, looking like sort of a miracle for an uninitiated topologist or algebraic geometer, one can compute exactly the combination of these orders leading to x (Mod/?). We refer to the original papers of Harer and Zagier [85] and Penner [199] for the details. While in [85] the needed technique (integration over the space of Hermitian matrices!) appears as a deus ex machine, Penner [199] motivates it by some ideas from quantum physics. A simplified proof was provided by Kontsevich [126]; see [126, Appendix D]. More recently, Zagier [243] provided a considerably simpler proof of the key combinatorial fact needed for the calculation of x(Mod/?). This new proof is based on the theory of characters of symmetric groups (and avoids the integration over the space of Hermitian matrices). Despite this dramatic simplification, the original proofs retain, in the author's view, considerable interest. D 6.7.B. B2g/4g(g - 1).

COROLLARY

If S is a closed surface of genus g ^ 2, then x(Mod5) =

The corollary immediately follows from the theorem if we use the short exact sequence PROOF.

1 ^ 7r 1 (5) ^ Mod5, -^ Mod^ -> 1, where ^i is the result of making one hole in S (cf. 2.8), the fact that virtual Euler characteristic is multiplicative in short exact sequences (cf. [26, Chapter IX, Proposition 7.3]), and the easy fact that x (^r i (5)) is equal to the Euler characteristic of S, i.e., to 2 — 2g. D Finally, we note that [85] also contains some information about the usual Euler characteristic E ( - l ) ' dim///(Mod5, Q). 6.8. Torsion in mapping class groups, torsion in their cohomology and related topics Mapping class groups contain many torsion elements, which can be exploited to detect torsion in their cohomology. Clearly, any finite group F acting on a surface S by orientationpreserving diffeomorphisms may be considered as a finite subgroup of Diff(5), and hence

596

N.VIvanov

leads to a finite subgroup of Mod^. In fact, the latter subgroup of Mod^ is isomorphic to F, as immediately follows from the next theorem. 6.8. A. Let S be an orientable surface of negative Euler characteristic. If F is a finite subgroup of T>\ff{S), then the canonical homomorphism

THEOREM

r^Aui{Hi(S,Z/mZ)) defined by the action of diffeomorphisms on homology is injectivefor any m ^ 3. Note that the homomorphism F -^ Aut(H\(S,Z/mZ)) obviously factors through Mod^. This theorem is essentially due to Serre [212]; for an elementary proof, see [108, Theorem 1.3 and Supplement 1.4]. According to a famous theorem of Kerckhoff [122,123], every finite subgroup of Mods can be obtained as the image of a finite subgroup of DiffC^); see Theorem 7.4.B. In practice finite subgroups of Mod^ are always obtained from an action of a finite group on 5, i.e., as images of finite subgroups of DiffC^). Kerckhoff's theorem tells us that we will not miss any finite subgroup by restricting ourselves to this method. Note that every finite group admits an embedding into some mapping class group because, as is well known, every finite group occurs as a group of symmetries of a Riemann surface. Moreover, Greenberg [66] proved that every finite group is isomorphic to the group of all automorphisms of some Riemann surface. Finite subgroups of Mod^ are used to detect nontrivial cohomology classes of Mod^ by restricting cohomology classes to appropriate finite subgroups. Of course, one can detect in this way only torsion cohomology classes, because cohomology groups of finite groups always consist of torsion classes. The first result of this sort was obtained by Chamey and Lee [35]. It was improved by Glover and Mislin [62,63] and by Charney and Lee [36]. It is convenient to state these results in terms of the stable cohomology groups in the sense of homology stability discussed in 6.5. Note that there are cohomology stability theorems exactly similar to the homology stability theorems of 6.5. In fact, stability theorems for cohomology follow from the corresponding stability theorems for homology. In particular, H^(A4s) does not depends on S if the genus of S is sufficiently large compared to m. Let us denote by H^(M) the group H^{Ms) for surfaces S of sufficiently large genus. Similar notation will be used for (untwisted) coefficients other than Z. 6.8.B. For all k the stable cohomology group H^^{M) contains an element of order equal to the denominator of the rational number B2k/'2k, where B2k is the 2k-th Bernoulli number THEOREM

This theorem is due to Glover and Mislin [62,63]. The numbers appearing in this theorem are the same as in the computation of the virtual Euler characteristic (cf. Theorem 6.7.A). This remarkable fact apparently has no (known) explanation. In order to construct elements of //*(A^), Glover and Mislin consider closed surfaces S and use the canonical representation Ms ^^ Aui(H\(S, C)) defined by the action of diffeomorphisms on homology. This leads to flat complex vector bundle (with the fiber H\ (S, C))

Mapping class groups

597

over K(Ais, 1), and the Chem classes of this bundle can be considered as elements of H*(Ms)' Since the bundle is flat, they are torsion elements. One needs to know that these Chem classes are non-zero and, moreover, have sufficientiy high order. This is checked by restricting (pulling back) them to specially constructed finite cyclic subgroups of A^^. The elements alluded to in the theorem are some combinations of these Chem classes. The cyclic subgroups used to detect the order of cohomology classes from Theorem 6.8.B are constmcted only for surfaces of fairly high genus (compared to the dimension of the classes). The (co)homology stability theorems imply that these cohomology classes survive till a much lower genus. It happens that sometimes these cyclic groups cannot act nontrivially on surfaces of such a low genus, and hence cannot be subgroups of the mapping class groups of these low genus surfaces. In this way, one gets examples of p-torsion cohomology classes of mapping class groups having no p-torsion elements (where p is some prime number). Such torsion elements in cohomology are somewhat unusual and are called exotic torsion elements. The existence of exotic torsion elements was observed in [63]. 6.8.C. The stable cohomology H'^iM, Z[l/2]) with coefficients Z[l/2] contains as a direct summand the cohomology H*(Im7, Z[l/2]) of the space \mJ with the same coefficients. THEOREM

This theorem is due to Chamey and Lee [36]. The space Im 7 is a well known character from homotopy theory. It is known that its cohomology groups consist of torsion elements; replacing the coefficients Z by Z[l/2] just kills the 2-torsion. Chamey and Lee use some cyclic subgroups in a manner similar to that of Glover and Mislin [63] (but the subgroups of Chamey and Lee are smaller and simpler to construct) together with some deep ideas from homotopy theory and algebraic ^-theory (the methods of Glover and Mislin are more elementary). Another difference is that Chamey and Lee use the canonical homomorphisms Ms ^^ Aut(//i {S, TJ/pTj)) (their images are contained in the symplectic groups over the field F^ = Z//7Z, and a lot is known about the cohomology of these groups from the work of Quillen and others) instead of the homomorphisms Ms -^ Aut(//i (5, C)) used by Glover and Mislin. Theorem 6.8.C can be used to produce some torsion classes in addition to those constructed by Glover and Mislin [63]. One can easily realize //*(A1) as the cohomology of some space. Let 5' be a surface of genus g with one boundary component. We may assume that S^ is contained in 5* , (and hence S^ .^ \ int 5^ is a toms with two holes) for all g > 1. Then we have a sequence of inclusions 5| ^^ • • • ^^ 5J -^ 5' j -^ - •, inducing inclusions of the corresponding mapping class groups A1} -> > M^a -^ - ^ l + i ~^ " ^ where M\ = AI51. Let M be the direct limit (i.e., the union) of these groups. Clearly, the cohomology H*(M) of this group is exactly what was denoted by Wi^M) above. By the definition, H'^(M) is the cohomology H'^iKiM, 1)) of the space K(M, 1). A natural problem is to refine Theorem 6.8.C and produce a sphtting on the level of spaces, not just cohomology groups. In fact, it is natural to replace K{M, 1) by the simply connected space K(M, 1)^ having the same cohomology groups as K(M, 1). Here ^ denotes Quillen's plus-construction, which is well-defined for K(M, 1) because H\ (K(M, 1)) = 0 as a corollary of Theorem 6.6.B.

598

N.VIvanov

Chamey and Cohen [34] proved that it is possible to find such a splitting in the stable homotopy category. Note that the difference between K(M, 1) and K(M, 1)^ disappears in the stable homotopy category. 6.8.D. (Im 7) 0 Z[l/2], /.^., the space Im J localized away from 2, is a stable retract of K {M, 1).

THEOREM

The proof uses some noncyclic finite groups 7tk (they are iterated wreath products of cyclic groups). But Charney and Cohen do not embed them in any Ms- Instead of this, they construct maps K{7Tk,\) ^^ K(M, 1)"^ (in fact, only over finite skeletons) using the homology stability theorems for mapping class groups in order to split natural maps of the form K{Ms, 1)^ ^^ K(MR, 1)^, where 5 is a surface with one boundary component and R is the result of gluing a disc to this boundary component. Recently, Tillmann [225] proved that the splitting of Theorem 6.8.C indeed can be realized on the level of spaces without passing to the stable homotopy category. Namely, she proved the following theorem. 6.8.E. There is a space Y such that K(M, l)^ is homotopy equivalent to ((Im7)(8)Z[l/2]) x F .

THEOREM

The proof is based on Theorem 6.8.D and the following fundamental theorem of Tillmann [225]. THEOREM

6.8.F. K(M, 1)"^ has the homotopy type of an infinite loop space.

This theorem is obviously of great interest independent of its applications to Theorem 6.8.E. The proof is based on a construction of suitable categories (using the results of a previous paper of Tillmann [224]) which lead to an acceptable input for an infinite loop space machine. Then a new group completion theorem and Harer's homology stability theorem (see 6.5) are used to identify the resulting infinite loop space with K{M, 1)^. Tillmann's infinite loop space structure on K{M, \)^ extends the loop space structure on K{M, 1)^ constructed previously by more elementary means by Miller [164] and Boedigheimer [22]. But it is not clear yet if it extends the double loop space structure of [164] and [22]. See also [42] for related results about the homology operations in 77*(At). Now we turn to the mapping class group Mod2 = Mod52 of a closed surface ^2 of genus 2. Lee and Weintraub [131] and Cohen [37] proved that //*(Mod2, Z) is a torsion group, having nontrivial p-torsion (for prime p) only for /? = 2, 3 or 5. Cohen [37,38] showed that the 5-torsion is completely captured by a cyclic subgroup of order 5. More precisely, the following theorem holds. THEOREM 6.8.G. There is a homomorphism 7J/5TJ -^ Mod2 inducing an isomorphism on 5-primary components in homology and on homology with coefficients Z/5Z. In particular, Hn (Mod2, Z) is equal to Z/5Z plus some 2- and 3-torsion groups for odd n and is a sum of 2- and 3-torsion groups for even n.

Mapping class groups

599

The cohomology of Mod2 with coefficients Z/2Z, Z/3Z, and Z/5Z was essentially completely described by Cohen and Benson [38,9,10]. For the coefficients Z/3Z and Z/5Z, even the ring structure of the cohomology was determined, while for the coefficients Z/2Z the ranks of the cohomology groups were computed, but the question of which of two possible ring structures is correct was left open. These papers also contain some information about the integral cohomology of Mod2, the structure of which was further elucidated by Cohen [39]. We state only the result for the coefficients Z[l/2], which is especially simple. THEOREM

6.8.H. The ring H*(Mod2, Z[l/2]) is isomorphic to

Z[l/2][x,y,z]/{5x,3y,3z,z^), where x, y, z have degrees 2, 4, 5 respectively. The proof invokes a connection between Mod2 and the mapping class group Modg Mod^6 with six holes S^ due to Birman and Hilden [19]. Namely, there is a Iodo6 of a sphere w short exact sequence 1 -> Z/2Z -> Mod2 -> Mod^ -> 1, with the image of Z/2Z generated by the hyperelliptic involution (cf. end of 4.3 for the latter). In addition to this, a close relation between ModQ and the Artin braid group on six strings is exploited. This geometric input leads to cohomological computations by means of some powerful homotopy-theoretic techniques. For further results obtained by this approach we refer to [40,41], where some noncyclic finite subgroups of A^^, namely quaternion and dihedral groups mMs, play an essential role. For further information about many of the topics discussed in this section, and for other related results, we refer to an excellent expository paper [167] by MisHn. In particular, it contains a discussion of results of Glover, Mislin and Xia [64,65], and of related results of Xia [235-239]; see also [240-242] and [207] for further results.

7. Thurston's theory and its applications This section is devoted to an overview of Thurston's theory of surface diffeomorphisms and some of its applications. For more details, we refer to Thurston's updated announcement of the main results [222], and for a detailed exposition to the proceedings of the Orsay seminar [53]. In the case of closed surfaces, a nice introduction from the point of view of hyperbolic geometry is provided by Casson and Bleiler [30]. See also Hatcher [90].

7.1. Classification of mapping classes An element / G Mod^ is called reducible if it fixes some simplex of C{S) (but, perhaps, permutes its vertices), and irreducible otherwise. As usual, we say that an element / is of

600

N.VIvanov

finite order if f^ = \ for some « 7^ 0. If an element / is not of finite order and is irreducible, then it is called pseudo-Anosov. It turns out that elements in all three classes (reducible, finite order, pseudo-Anosov) can be represented by diffeomorphisms having some special properties. These special representatives play a crucial role in the understanding of corresponding mapping classes. First of all, for a reducible element / fixing a simplex a of C{S), let us consider the union C of some disjoint circles representing the vertices of a. We will call such a union a realization of a. In this case there is a diffeomorphism F \S ^^ S representing the isotopy class / and leaving C invariant; F{C) = C. Such a diffeomorphism F induces a diffeomorphism Fc'Sc -^ Sc of the result Sc of cutting S along C. The components of Sc are, in a definite sense, simpler than S (they have larger Euler characteristic) and this allows the use this procedure in inductive arguments. One minor difficulty comes from the fact that Sc has, in general, several components, and Fc may permute them in a nontrivial way. The standard way to deal with this is to replace Fc by some power of it not permuting the components. Another, more serious, problem comes from the fact that Fc may be isotopic to the identity, and then we lose all the information after the cutting. This happens exactly when / is a product of Dehn twists around the components of C. This case needs to be dealt with separately. Anyhow, any reducible mapping class can be represented by a reducible diffeomorphism, i.e., a diffeomorphism preserving the union of several pairwise disjoint and non-isotopic nontrivial circles. While the results outlined in the previous paragraph are essentially elementary, the next case, namely the case of elements of finite order, is much deeper. Special representatives of such elements are provided by the following theorem of Nielsen [194]. 7.1. A. Iff^ Mods is an element offiniteorder n^O, then f can represented by a diffeomorphism F : S ^ S of the same order n. Moreover, one can choose F to be an isometry of a metric of constant curvature on S with geodesic boundary. THEOREM

See [53, Expose 11, §4] for a proof of this theorem in the context of Thurston's theory and an outline of another proof due to Fenchel [54]. As a simple application of this theorem, consider the kernel Fsim) of the natural homomorphism Mod5 -> Aut{Hi(S, Z/mZ)) defined by the action of diffeomorphisms on homology. Clearly, Fs{m) has finite index in Mod^. COROLLARY

7.1 . B . Ifm ^ 3, then Fs(m) is torsion free.

PROOF. If the Euler characteristic of S is negative, it is sufficient to combine Nielsen's Theorem 7.LA with Serre's Theorem 6.8.A. The remaining cases are elementary and left to the reader. D

Finally, the remaining elements, which we called pseudo-Anosov, also can be represented by some very special diffeomorphisms, namely hy pseudo-Anosov diffeomorphisms.

Mapping class groups

601

(To be honest, they are only homeomorphisms, but their non-smoothness is very mild and harmless.) We will not give their definition here, which generalizes (the two-dimensional case of) the definition of Anosov dijfeomorphisms, introduced by Anosov [2] and playing a central role in the theory of dynamical systems; see [53] or [222]. The existence of pseudo-Anosov diffeomorphisms is one of the main achievements of Thurston's theory of the diffeomorphisms of surfaces. While the existence of nontrivial reducible mapping classes or mapping classes of finite order is quite obvious, it is not so for pseudo-Anosov mapping classes. Let us indicate a couple of ways to get such mapping classes. We need the following notion: we say that a set A of vertices of C{S) (i.e., a set of isotopy classes of nontrivial circles on S) fills S if for every vertex y there is a vertex a ^ A such that any two circles representing of, y have a non-empty intersection. For finite sets A, this is equivalent to the following property: for any collection of circles representing all isotopy classes from A, all components of the complement of the union of these circles in S are either (open) discs or (half-open) annuli, and each of the latter contains a component of 95. THEOREM 7.I.C. Ifa,p are two isotopy classes of nontrivial circles such that (of, P) fills S, then all elements t^tg"^ form, n ^ I are pseudo-Anosov.

See [53], Expose 13, Remark 2 after Theorem III.3. Theorem III.3 itself shows that under the assumptions of Theorem 7.1 .C, many other combinations of ta, t^ are also pseudoAnosov. These results were generalized by Long [138] and then by Penner [200] to the situations when more than two Dehn twists are involved. Further examples were provided by Bauer [7]. The following remarkable result, due to Fathi [52], while not giving completely explicit examples of pseudo-Anosov elements, shows that they are in a very strong sense generic. THEOREM 7.I.D. Let f e Mods and a be a vertex ofC(S). If the set {/"(of): n eZ} fills S, then all elements t^f are pseudo-Anosov except for at most 1 consecutive values ofn.

For example, if {a, fi} fills 5, then {tUa): n eZ} also fills S, The paper [52] includes some other results of this sort, but the following question remains open: is it possible to replace the Dehn twist ta in this theorem by an arbitrary composition of Dehn twists about several disjoint circles, possibly at the cost of replacing the number 7 by some, may be undetermined, but universal number? The results of Fathi were preceded by a theorem of Long and Morton [140] in which only the finiteness of the number of exceptions (i.e., number of n such that t^ f is not pseudo-Anosov) was asserted, instead of the existence of a universal constant (^ 7 by Theorem 7.1 .D). When one deals with individual mapping classes, one can usually reduce the problem to the case of irreducible mapping classes by the method of cutting outlined above. When one deals with the whole mapping class group or a general subgroup of it, usually there is no way to avoid reducible elements. But one can often restrict attention to following, in some sense simplest, type of reducible elements. Let us call a mapping class / G Mod^ pure if it can be represented by a diffeomorphism F: S -^ S fixing (pointwise) some union C

602

N.VIvanov

of disjoint and pairwise non-isotopic nontrivial circles on S and such that F does not permute the components of 5 \ C and induces on each component of the cut surface Sc a diffeomorphism isotopic either to a pseudo-Anosov or to the identity diffeomorphism. For example, every pseudo-Anosov element is pure, as is also any product of Dehn twists about several disjoint circles. In the latter case the induced diffeomorphism of Sc is isotopic to the identity; in all other cases F induces a diffeomorphism isotopic to a pseudo-Anosov diffeomorphism on at least one component of Sc • THEOREM

7.1 . E . Ifm^3,

then Fsim) consists of pure elements.

This theorem strengthens Corollary 7.1 .B, and is a natural sharpening of it in the framework of Thurston's theory. For a proof, see [108, Corollary 1.8]. This theorem often allows the immediate elimination of all difficulties related to elements of finite order and to possible permutations of components (of the cut surface) by reducible diffeomorphisms. For example, if one considers a subgroup G of Mod^, it is often possible to replace it by Another important tool for dealing with reducible elements is the notion of essential reduction classes introduced, in a slightly different form, by Birman, Lubotzky and McCarthy [20]. For a pure element / G Mod^ we say that a vertex a of C{S) is an essential reduction class of / if f {pi) = a and f{^) ^ P for all P such that a, fi cannot be represented by disjoint circles. For a general element / e Mod^, we say that a vertex a of C(S) is an essential reduction class of / if it is an essential reduction class of some nontrivial (and than, as one may prove, of any nontrivial) power / " (n ^ 0), which is pure. Note that / has pure powers in view of Theorem 7.1.E and the fact that Fsim) has finite index in Mod^. It turns out that the set of all essential reduction classes of / is a simplex of C(5'), which we call the canonical reduction system of / and denote by a ( / ) . If / is pure, it fixes all vertices of or(/). In general / leaves cr{f) invariant. The crucial fact is that a (/) is non empty if / is a reducible element of infinite order ( a ( / ) is empty for any element of finite order / , reducible or not). In particular, cutting along some realization of cr(f) provides a canonical way to simplify a reducible element / . If / is pure, then this leads to a diffeomorphism (of the cut surface) which leaves all components invariant and is isotopic on any component either to the identity, or to a pseudo-Anosov diffeomorphism.

7.2. Measuring against circles and its applications In this section we discuss one of the basic technical tools of Thurston's theory: measuring various geometric objects on a surface S against the isotopy classes of nontrivial circles, i.e., the vertices of C{S). We will denote by V(S) the set of all vertices of C(5'), i.e., the set of all isotopy classes of nontrivial circles. One of the fundamental ideas of Thurston was to consider measurements with respect to all vertices simultaneously. The natural place to record all such measurements is the set 71{S) of all functions V(S) -^ R^o. endowed with the product topology. The mapping class group Mod^ acts on V (5), and this action induces an action of Mod^ on 1Z(S). The next important idea is to pass from the space Tl(S) to a projectivization of it, namely to the quotient V7l(S) of TliS) \ {0} by the natural (diagonal)

Mapping class groups

603

action of the multiplicative group R>o on it. Clearly, the action of Mod^ on 1Z(S) induces an action of Mod^ on V1Z(S). Let us illustrate this approach by the example of the Teichmuller space Ts. A natural (but not the only possible) way to measure the points of Ts (i.e., the classes of hyperbolic metrics on S) with respect to the vertices a e V(S) is to use the length functions la'-Ts -> R>o from 5.2. The collection of all length functions defines an embedding Ts -^ 1Z(S). Clearly, this embedding is Mod^-equivariant. Its image is contained, obviously, in 7l(S) \ {0}. The composition Ts -^ V7l(S) of this embedding with the canonical projection 1Z(S) \ {0} -^ V1Z{S) is still an embedding, which is also Mod^-equivariant. It turns out that the embedding Ts -^ V1Z{S) is a homeomorphism onto its image, this image has a compact closure in V7Z(S) and, moreover, this closure is homeomorphic to a closed disc (of the dimension equal to the dimension of Ts) under a homeomorphism taking the image of 7^ into the interior of this disc. In this way, we achieve a natural compactification of T^ by a sphere called Thurston's boundary of Teichmiiller space. The compactification itself is called Thurston's compactification of Teichmuller space. An advantage of this compactification is that the action of Mod^ extends to it by continuity (as is clear from the construction). Kerckhoff [121] proved that this desirable property does not hold for the classical compactifications of Teichmuller space by a sphere, the so-called Teichmuller compactifications, A Teichmuller compactification depends on a choice of a base point in Ts, and Kerckhoff's result holds for any choice of the base point. In fact, Kerckhoff proved that compactifications corresponding to different base points are often different, and this is closely related to the impossibility to extend the action of Mod^ to any of them by continuity. Another example of the idea of measuring geometric objects against the isotopy classes of circles is provided by measuring the isotopy classes of circles themselves. For a, P e V(S) let us define the geometric intersection number i(Qf, P) as the minimal number of points in the intersections A (1 B, where A, B run over all circle representatives of the isotopy classes a, P respectively. For any a eV(S) the function I^ : y \-> iiy, a) belongs to 7l(S). Moreover, it is easy to see that for any a eV(S) there is some y eV(S) such that i(Qf, y) ^ 0. Hence, /Q, G ^(S) \ {0} for every a eV(S).By assigning /c^ to of, we get a map V(S) -^ ^(S) \ {0}. It is easy to see that this map is injective. Moreover, its composition with the projection 11(5) \ {0} -> V7Z(S) is still injective (this is a little more difficult). The remarkable fact is that the image of V(S) under this composition V(S) -^ V1Z{S) is contained in Thurston's boundary of Ts and is dense in this boundary. In some sense, this image plays the role of rational points of Thurston's boundary. It turns out that other points of Thurston's boundary also admit a geometric interpretation. The relevant geometric object is a measured foliation with singularities, or, more precisely, the Whitehead equivalence class of such a foliation on S. Usually one speaks simply about measured foliations on S. The set of all (Whitehead equivalence classes) of measured foliations on S is denoted by MT{S). We omit the definitions, referring to [53] and [222], and restrict ourselves to the following remarks. A measured foliation on 5* is a codimension 1 foHation on S (so the leaves have dimension 1). Since the Euler characteristic of S is almost never zero, S usually does not admit honest foliations, and we have to allow some singularities in them. The term measured means that our foliation is equipped with a so-called transverse measure, which allows us to measure locally (say, within a flow

604

N.VIvanov

box of the foliation) the distance between the leaves. The transverse measure allows us to define the transverse length of curves and, in particular, of circles (if a curve is contained in a leaf, its transverse length is 0). The notion of the transverse length of a circle allows us to deal with the measured foliations in a way similar to the way we dealt with hyperbolic metrics (or, rather, points of Ts) and circles (or, rather, vertices of C{S)) above. Namely, given a vertex y eV(S) and a measured foliation /x, we define ly (/x) to be the infimum of transverse lengths with respect to /x of all circles in S representing the isotopy class y. In contrast with the hyperbolic metrics (compare 5.2), this infimum is not always achieved (but it is achieved on a not necessarily embedded curve homotopic to circles representing y). For any measured foliation /x, the function L^:y \-^ ly (/x) belongs to 1Z(S). It turns out that for every measured foHation /x there exists some y eV(S) such that ly (JJL) 7^ 0. Hence, L^ e 1Z(S) \ {0} for every /x e MT{S). By assigning L^ to /x, we get a map MJ^{S) -^ 71(3) \ {0}. It is not very hard to see that this map is injective. In particular, we can equip MT{S) with the topology induced from 1Z{S) by this map. The composition of this map with the projection 7l(S) \ {0} -^ VIZ(S) turns out to be not injective. The reason is very simple: the multipHcation of transverse measures by positive constants defines a natural action of the multiplicative group R>o on MJ^(S), and our map MT{S) -^ Tl{S) \ {0} is obviously R>o-equivariant. So, it is only natural to consider the quotient VMT{S) of MJ^iS) by this action of R>o. Clearly, our map MT{S) -^ 11(3) \ {0} leads to a map VMJ-'(3) -^ V1Z(3), which turns out to be injective. The image of this map is equal to Thurston's boundary of Ts. The interpretation of points of Thurston's boundary in terms of measured foliations plays a crucial role in the proof of the fact that Thurston's boundary is a sphere forming a ball together with the Teichmiiller space. Note that any isotopy class of a nontrivial circle naturally leads to an equivalence class of measured foliations, and the images mlZ(3) of an isotopy class of a circle and of the corresponding measured foliation are the same. So, a measured foliation turns out to be, in some sense, a generalized circle. It is worth pointing out that Thurston's boundary can be constructed and its main properties can be established, using only the notion of a measured foliation without any reference to Teichmiiller spaces, and for many applications it is sufficient to use only Thurston's boundary itself (together with the action of Mod^ on it). 7.3. Action of the mapping classes on Thurston's compactification of Teichmiiller space Let us denote by Ts Thurston's compactification of Teichmiiller space, i.e., the union of Ts with its Thurston's boundary. In this section we describe the main features of the action of various classes of elements of Mods introduced in 7.1 on 7^. First of all, the elements of finite order can be characterized by the fact that they have a fixed point in the Teichmiiller space Ts itself. The set of fixed points in Ts can be identified with some other Teichmiiller space. A similar result probably holds for the set of fixed points in r ^ . _ A pseudo-Anosov element / has exactly twofixedpoints in Ts and both of them belong to Thurston's boundary; there are no fixed points in Ts itself. One of the fixed points is in a

Mapping class groups

605

natural sense attracting, and the other one is repelling. In the following theorem a denotes the attracting point and r denotes the repelling point. 7.3.A. Let f be a pseudo-Anosov element. There exist two points a, r in Thurston's boundary of Tg, such that if U is any neighborhood of a in Ts and K is any compact set in Ts \ {r}, then f^(K) C U for all sufficiently large n. THEOREM

So, all points in Ts\{r} converge under the action of the iterations of f to a, uniformly on compact sets. This justifies the terms "attracting" and "repelling". Since both Ts and its boundary are obviously invariant, a similar result holds for Thurston's boundary instead of r ^ . One can find a somewhat weaker result (not asserting the uniformity of the convergence) in [53] (cf. [53, Expose 12, Corollaire II. 1]); Theorem 7.3.A itself easily follows, for example, from Theorem 7.3.B below. The action of reducible elements is more complicated. If a reducible pure element / is not a composition of Dehn twists about several disjoint circles (so, if we cut S along a realization of a ( / ) , then the induced diffeomorphism of the cut surface is isotopic to a pseudo-Anosov diffeomorphism on at least one component), then we have a picture similar to the above description of the action of a pseudo-Anosov element. We state the result only for the action on Thurston's boundary. Its set of fixed points consists of two components, one of which is in a natural sense attracting, while the other one is repelling. In the following theorem, A denotes the attracting set, and R denotes the repelling set. 7.3.B. Let f be a pure element of Mods which is not a composition of Dehn twists about several disjoint circles. There exist two disjoint subsets A, R of Thurston's boundary ofTs, such that ifU is any neighborhood of A in Thurston's boundary and K is any compact set in the complement of R, then / " (K) C U for all sufficiently large n. THEOREM

This result is proved in [108, Appendix], where also the components of the fixed point set are completely described. Theorem 7.3.A is a special case of Theorem 7.3.B and this description. Note that a weaker result, in which the attracting and the repelling sets are not disjoint in general (cf. [108, Theorem 3.5]), is sufficient for most applications. Another version of this weaker result was proved by McCarthy [159]; cf. [159, Uniform Convergence Lemma].

7.4. Some applications We start with a famous result of Thurston about 3-manifolds, which in fact served as the motivation for Thurston's theory of surface diffeomorphisms. Recall that the mapping torus Mf of map / : X ^ X is the quotient space X x [0, l]/(x, 0) ~ (/(jc), 1). If X is a manifold and / is diffeomorphism, then Mf is also a manifold. THEOREM 7.4. A. Let f :S ^^ S be a diffeomorphism of a closed surface S. The mapping torus Mf admits a Riemannian metric of constant negative curvature if and only if the isotopy class off is pseudo-Anosov.

606

A^. V Ivanov

A slightly more complicated result holds for surfaces with boundary as well. This theorem is the most difficult special case of Thurston's famous geometrization theorem for Haken manifolds. Of course, the proof involves much more than Thurston's theory of surface diffeomorphisms. The first account of this theorem is due to Sullivan [220]; the exposition in [220] is very demanding. Recently, new expositions of this theorem were published by McMuUen [162] (see [162, Chapter 3]) and Otal [195]. McMuUen's approach is close to the original ideas of Thurston (and refers to some still unpublished preprints of Thurston). The approach of Otal follows the one of Thurston in the general outline, but provides a completely new proof of one of the key steps (the so-called double limit theorem); his exposition is essentially self-contained. Hopefully, Thurston's own proof [223] will also appear soon. See also notes by Kapovich [120]. The next theorem, due to Kerckhoff [122,123], provides a solution of the so-called Nielsen realization problem. THEOREM 7.4.B. Let G be afinitesubgroup of Mods. Then there is afinitesubgroup G^ of Diff(5) which maps isomorphically onto G under the natural map Diff(5') -> Mod^. Moreover, one can choose G^ to be a subgroup of the group of isometrics of some metric of constant curvature on S with geodesic boundary.

One says that G^ realizes G. Note that this theorem generahzes Theorem 7.LA from cyclic to arbitrary finite subgroups of Mod^. The key ingredient of the proof is provided by the so-called earthquake flows on the Teichmiiller space Ts. These flows are related to measured foliations in the same manner as the Fenchel-Nielsen flows from 5.2 are related to circles. In fact, earthquake flows are the limits of (suitably chosen and parameterized) Fenchel-Nielsen flows. The main properties of the earthquake flows used in the proof of Theorem 7.4.B are the convexity of the length functions from 5.2 along the flow lines of earthquake flows and the fact that every two points can be connected by the flow line of an earthquake flow. We refer to [123] for the details; see also [124]. Some further developments resulting from this solution of the Nielsen realization problem are presented in [125]. After Kerckhoff's work, Wolpert [234] and Tromba [228] (see [228, Section 6.4]), [229] found other proofs of Theorem 7.4.B, independent of Thurston's theory (but still based on convexity properties of some functions; in fact, the same length functions in Wolpert's proof, but along different paths in T^). The first applications of Thurston's theory to the algebraic properties of mapping class groups are due to McCarthy [158], who noticed that Thurston's theory provides a good description of normalizers and centralizers of pseudo-Anosov elements, and to Birman, Lubotzky and McCarthy [20], who proved, in particular, the following theorem. 7.4.C. If a subgroup of Mod^ contains a solvable subgroup of finite index, then it contains an Abelian subgroup of finite index.

THEOREM

Soon after [20], the author proved [95] the following more strong theorem. 7.4.D. Let G be a subgroup o/Mod^, and let m > 3. Then either G contains a free (non-Abelian) group with two generators, or G D Fsim) is Abelian. THEOREM

Mapping class groups

607

Since Fsim) is of finite index in Mod^ and a solvable group cannot contain a free group with two generators, Theorem 7.4.D implies Theorem 7.4.C. The following corollary of Theorem 7.4.D, which is only slightly weaker than Theorem 7.4.D itself, was simultaneously and independently proved by McCarthy [159]. THEOREM 7.4.E. Let G be a subgroup of Mods. Then either G contains a free group with two generators, or G contains an Abelian subgroup of finite index.

A similar property, with "Abelian" replaced by "solvable", holds for arbitrary linear groups (i.e., subgroups of GL„ (k) for a field k) according to a famous theorem of Tits [226]. In fact, the search for an analogue of Tits' theorem for subgroups of the mapping class groups led to Theorems 7.4.D and 7.4.E. Later on, the author [95,99] discovered that some other fundamental theorems about the linear groups have analogues for subgroups of mapping class groups. In order to state them, recall that a subgroup / / of a group G is called maximal if for every subgroup ^ of G such that H C K, either H = K or K = G. Recall also that the Frattini subgroup F(G) of a group G is defined as the intersection of all maximal subgroups of G. An intuitive characterization of F(G) is provided by the following elementary result: F(G) is the set of all nongenerating elements of G, i.e., such elements g e G such that for every generating set X of G, the set X \ {g} is also generating. 7.4.F. Let G be a finitely generated subgroup of Mod^. Then either G contains a maximal subgroup of infinite index, or G contains an Abelian subgroup of finite index.

THEOREM

It is easy to see that if a group G contains an Abelian subgroup of finite index, then all maximal subgroups have finite index. While usually it is very easy to find a maximal subgroup of finite index (at least when there are nontrivial subgroups of finite index: take a subgroup of minimal index 7^ 1), it is hard to prove the existence of maximal subgroups of infinite index. For linear groups, a theorem similar to Theorem 7.4.F (with "solvable" instead of "Abelian") was proved by Margulis and Soifer [151]; their result motivated Theorem 7.4.F. THEOREM 7.4.G. Let G be a finitely generated subgroup of Mod^. Then the Frattini subgroup F(G) is nilpotent.

This is an analogue of a theorem of Platonov [203] to the effect that Frattini subgroups of finitely generated linear groups are nilpotent. It was proved by the author [99] after some partial results of Long [139]. The proofs of Theorems 7.4.C-7.4.G are based on the properties of the action of the elements of Mod^ on Thurston's boundary of the Teichmiiller space Ts and, in particular, on the results discussed in 7.3. A crucial role is also played by the following result [97,99], in which we call a subgroup G of Mod^ irreducible if there is no nonempty simplex a of C{S) such that g{a) = o for all g G G. THEOREM 7.4.H. If G is an infinite irreducible subgroup of Mod^, then G contains a pseudo-Anosov {in particular, an irreducible) element.

608

N.VIvanov

Note that a similar statement for finite subgroups is false: according to Oilman [61], there are finite irreducible subgroups of Mod^ consisting entirely of reducible elements. In view of Theorems 7.4.B and 7.4.H, it is natural to classify the subgroups of Mod^ in a manner similar to Thurston's classification of elements of Mod^. Namely, a subgroup of Mod^ can be reducible (i.e., not irreducible), finite, or infinite irreducible. (Note that these classes are not completely mutually exclusive: a subgroup can be finite and reducible.) The following theorem [108] provides a useful complement to Theorem 7.4.H and this classification. 7.4.1. If a subgroup G of Mod^ contains a pseudo-Anosov element, then either G contains an infinite cyclic subgroup of finite index generated by a pseudo-Anosov element, or G contains a free group freely generated by two pseudo-Anosov elements. THEOREM

Theorems 7.4.C-7.4.G are proved using the same general outline. First, we replace our group G by G n Fsim) for some m > 3. If the resulting subgroup (still denoted by G) is reducible, then one may cut S along some realization of a nonempty simplex a of C(S) such that g{a) = G for all g G G. This reduces the problem to a similar problem on the cut surface (which has simpler components). If a subgroup G is irreducible and infinite, then Theorems 7.4.H and 7.4.1 provide us with pseudo-Anosov elements in G, which play the key role in the rest of the proof. (Note that finite groups present no problems in these theorems.) For the details, see [108]. 7.5. Commuting elements and Dehn twists Thurston's theory allows us to formulate a very useful criterion for two elements of Mod^ to commute. We state it only for the pure elements, this case usually being sufficient for the applications. 7.5 .A. Let / , g be two pure elements of Mod^. These elements commute (i.e., fg = gf) if and only if there is simplex o ofC(S), its realization C C S and representatives F,G : S ^ S of f, g respectively such that: (i) F and G preserve (setwise) every component ofC and every component ofS\C; (ii) for every component R of the result Sc of cutting S along C, the diffeomorphisms FR,GR: R ^^ R induced on R by F,G respectively are isotopic to some powers of the same pseudo-Anosov diffeomorphism of R or to the identity. THEOREM

The proof is based on the results of 7.3; see [108, Section 8.12]. As an application, we give a purely algebraic characterization of powers of Dehn twists inside subgroups of pure elements. Recall that the centralizer CG(g) of an element g of a group G is defined as the subgroup {/ e G: fg = gf} and that the center C(G) of a group G is defined as the subgroup {/ € G: fg = gf for all g € G}. 7.5.B. Let G be a subgroup of Mods consisting entirely ofpure elements (for example, let G C Fs(m) for some m ^ 3). An element g e G is a nontrivial power of a Dehn twist if and only if the center C(CG(g)) of the centralizer CG(g) of g is isomorphic to Z and is not equal to the centralizer Co (g) of g.

THEOREM

Mapping class groups

609

In fact, CiCcig)) is isomorphic to Z only for nontrivial powers of Dehn twists and for pseudo-Anosov elements. The second condition serves to exclude pseudo-Anosov elements. This theorem is a version of a characterization of Dehn twists in [101]. Since powers of Dehn twists obviously are pure elements, one can apply Theorem 7.5.A to them and get the following result (admittedly, known before Thurston's theory). 7.5.C. Let ta, t^ be two Dehn twists and let m, n be two nonzero integers. Then t^t^ = t^t^ if and only if a and P can be represented by disjoint circles.

THEOREM

Theorem 7.5.A also allows us to easily find centers of the mapping class groups. THEOREM 7.5.D. If S is neither a sphere with < 4 holes, nor a torus with ^ 2 holes, nor a closed surface of genus 2, then the center CCMod^) is trivial (i.e., CCMod^) = 1). If S is a closed surface of genus 2, then the center CCMod^) is isomorphic to Z/2Z and is generated by the hyperelliptic involution from 4.3.

The corresponding result for the cases excluded in this theorem is only slightly more complicated. Theorem 7.5.D is well known, but apparently there is no good reference for it. COROLLARY 7.5.E. If S is neither a sphere with ^ 4 holes, nor a torus with ^ 2 holes, nor a closed surface of genus 2, then the canonical action o/Mod^ on C(S) is effective. If S is a closed surface of genus 2, the kernel of the canonical action o/Mod^ on C(S) is equal to the subgroup of order 2 generated by the hyperelliptic involution from 4.3.

This easily follows from Theorem 7.5.D, if one uses the fact that Dehn twists generate PMod^ (by Theorem 4.2.C).

8. Automorphisms of complexes of curves 8.1. The theorem about automorphisms The action of Mod^ on C{S) induces a homomorphism Mod^ -^ Aut(C(5')). This homomorphism is injective if S is neither a sphere with ^ 4 holes, nor a torus with ^ 2 holes, nor a closed surface of genus 2; cf. Corollary 7.5.E. It turns out that this homomorphism is also surjective in these cases, and also for closed surfaces of genus 2. THEOREM 8.1. A. Suppose that S is not a sphere with ^ 4 holes and not a torus with ^ 2 holes. Then all automorphisms ofC(S) are induced by the elements o/Mod^. In particular, if in addition, S is not a closed surface of genus 2, then Aut(C(5')) is canonically isomorphic to Mod^. If S is a closed surface of genus 2, then Aut(C(5)) is canonically isomorphic to the quotient of the group Mod^ by the subgroup of order 2 generated by the hyperelliptic involution.

610

N.VIvanov

This theorem is due to the author [103] for surfaces S of genus ^ 2, and to Korkmaz [127] in the remaining cases. See also [110] for an expanded and updated version of [103]. Most of the proof (namely, the ideas discussed in 8.4) for the genus > 2 case works also in the genus 0 and 1 cases. The main difficulty in treating these low genus cases, overcome by Korkmaz, consisted in finding and proving an analogue of Lemma 8.2. A below. If S is either a sphere with 4 holes, or a torus with ^ 1 holes, then C(S) is an infinite set of vertices without any edges (i.e., dimC(5) = 0) and, hence, Aut(C(5)) is an infinite symmetric group. Obviously, diffeomorphisms of S cannot permute vertices of C(S) arbitrarily, even in these cases, and thus the conclusion of the theorem is not true for such S. If 5 is a sphere with at most 3 holes, then C(S) is empty, but Mod^ is not trivial (it includes at least one orientation-reversing mapping class). Hence, the conclusion of the theorem is not true, by trivial reasons, for such S also. Luo [147] observed that Aut(C(5)) is not equal to Mod^ if 5 is a torus with 2 holes. The reason is very simple: If 5i ,2 is a torus with 2 holes and So,5 is a sphere with 5 holes, then C(5i,2) is isomorphic to C(5o,5), but Mod^, 2 is not isomorphic to Mod^^^g; in fact, Mod^^^ acts transitively on the set of vertices of C(5o,5) and the action of Mod^, 2 on the set of vertices of C(5i,2) has two orbits, corresponding to separating and nonseparating circles (cf. 8.6 for a similar phenomenon). Luo [147] also suggested another approach to Theorem 8.1.A, still based on the ideas of [103] (outlined below) and also on a multiplicative structure on the set of vertices of C{S) introduced in [145]. Theorem 8.1 .A is similar to, and was motivated by, a well known theorem of Tits [227] to the effect that all automorphisms of a Tits building (with some exceptions, as in our case) stem from automorphisms of the corresponding algebraic group; see [227, Introduction, Problem (B)]. The theorem of Tits, in turn, has its roots in the Fundamental Theorem of Projective Geometry, asserting that maps of a projective space to itself which map lines to lines, planes to planes etc., are, in fact, (projectively) linear. In 8.2-8.4 we will outline the main ideas of the proof of Theorem 8.1.A. After this, we will discuss applications of this result to mapping class groups and Teichmiiller spaces.

8.2. Intersection number 1 property We start by introducing some terminology. Two isotopy classes a, y6 of non-trivial circles on S (i.e., two vertices of C{S)) are said to have geometric intersection number 1 (respectively, 0) if they can represented by two circles intersecting transversely in exactly one point (respectively, by two disjoint circles). If this is the case, we write i(a, ^) = 1 (respectively, i(Qf, ^6) = 0). (These definitions form a special case of a general notion of geometric intersection number from 7.2.) Clearly, two isotopy classes have geometric intersection number 0 if and only if they are connected by an edge in C{S). Obviously, these intersection numbers are preserved by the action of Mod^. The starting point of the proof of Theorem 8.1. A (at least for surfaces of genus > 1) is the fact that the property of having geometric intersection number 1 is preserved by all automorphisms of C(S) (which a priori preserve only the property of having geometric intersection number 0). The key part of the proof of this fact is contained in the following lemma, at least when the genus is ^ 2.

Mapping class groups

611

Fig. 14.

LEMMA 8.2.A. Suppose that the genus of S is at least 2. Let oi\, a^ he isotopy classes of two nontrivial circles on S. Then the geometric intersection number!(«i, 012) = 1 if and only if there exist isotopy classes 0^3, 0^4, a^ of nontrivial circles having the following two properties: (i) i(Qf/, Qfy) = 0 if and only if the i-th and j-th circles in Figure 14 are disjoint, (ii) if Qf4 is the isotopy class of a circle C4, then C4 divides S into two parts, one of which is a torus with one hole containing some representatives of the isotopy classes Oil,

Oil-

PROOF. Figure 14 provides a proof of the "only if" part. In view of the property (ii), the proof of the "if" part can be reduced to an examination of a torus with one hole, which is not difficult. We omit the details. D

8.2.B. Suppose that the genus of S is at least 2. Every automorphism of C(S) maps pairs of vertices having geometric intersection number 1 into pairs of vertices also having this property. COROLLARY

It is sufficient to show that the properties (i) and (ii) of Lemma 8.2.A are preserved by every automorphism of C(5). As we noted before the lemma, two vertices have geometric intersection number 0 if and only if they are connected by an edge in C(5'). It follows that the property (i) is preserved by all automorphisms. In order to prove that the property (ii) is also preserved, we will show how to express it in terms of the structure of C(S) only. Let us consider a vertex a = (C) of C(S). Let La be the link of a in C(S), i.e., the set of all vertices of CiS) connected by an edge with a (with the induced structure of a simplicial complex). Consider the graph L*, having the same vertices as the link L^ and having as edges exactly those pairs of vertices which are not edges of La. It is easy to see that the connected components of L* correspond exactly to the connected components of the result Sc of cutting S along C (at least if none of them is a sphere with 3 holes). PROOF.

612

N.V Ivanov

Fig. 15.

After detecting the components of Sc with the help of L* (when one of the components is a sphere with 3 holes, we can detect only the other, but this turns out to be sufficient), we can return to the link La itself and try use Corollary 3.3.B, or the table preceding it, to recognize the topological type of these components. Let us apply these remarks to a^. The part of S not containing representatives of ofi, a2 should have genus one less than the genus of S and one more boundary component. If the genus of 5 is ^ 3, these properties can be recognized with the help of Corollary 3.3.B. The case of genus 2 is only a little more difficult; one should use not only this corollary, but also the table preceding it. It follows that the property (ii) can be expressed in terms of the structure of C(S) alone and, hence, is preserved by all automorphisms of C{S). This completes the proof. D Now, we turn to the genus 1 case. The following analogue of Lemma 8.2.A was proved byKorkmaz[127]. LEMMA 8.2.C. Suppose that S is a torus with at least two holes. Let a\, 0L2 be isotopy classes of two nontrivial circles on S. Then the geometric intersection number i(a\, Qf2) = 1 if and only if there exist isotopy classes 0^3, Qf4, 0^5 having the following two properties: (i) i(Qf/ ,aj) = 0 if and only if the i-th and j-th circles in Figure 15 are disjoint; (ii) if at is the isotopy class of a circle C/, then the circles C\, C2 and C3 are nonseparating, and both C4 and C5 divide S into a torus with one hole and a sphere with b-\-l holes, where b is the number of boundary components ofS. PROOF. Similarly to the proof of Lemma 8.2, the proof of the"only if" part is provided by the Figure 15. The proof of the "if" part is a little more difficult than the proof of the "if" part of Lemma 8.2, but we again omit the details. D COROLLARY 8.2.D. Suppose that S is a torus with at least two holes. Every automorphism ofC(S) maps pairs of vertices having geometric intersection number 1 into pairs of vertices also having this property.

Mapping class groups

613

1

Fig. 16.

The proof is more difficult than that of Corollary 8.2.B, partly because the table preceding Corollary 3.3.B is not sufficient to distinguish between the surfaces of genus 0 and 1. We omit fairly complicated details. Surprisingly, the vertices a\,.. .,as of both Lemmas 8.2.A and 8.2.C with the edges connecting them form a pentagon, presented in Figure 16, and a similar pentagon plays a role in the case of genus 0.

8.3. The case of a sphere with holes At first sight, none of the above ideas can be applied when 5 is a sphere with holes: all circles on a sphere with holes are separating and, hence, two isotopy classes never have geometric intersection number 1. It turns out that instead of non-separating circles, one can use circles bounding a disc with two holes in S. Then the role of pairs of isotopy classes having geometric intersection number 1 is taken over by the pairs of isotopy classes of such circles having the simplest possible non-trivial intersection, namely, intersecting as in Figure 17, where the shadowed discs represent holes. Such pairs of isotopy classes can be characterized in a manner similar to Lemmas 8.2.A and 8.2.C. A pentagon as in Figure 16 appears again, but its five vertices are not sufficient now. We refer to [127] for the details.

Fig. 17.

614

N.VIvanov

Note that the circles bounding discs with two holes and pairs illustrated in Figure 17 are exactly what is needed to make the machinery of the next section work (cf. the discussion of codings in 8.4).

8.4. Complexes of curves and ideal triangulations Now we describe the ideas involved in the remaining part of the proof of Theorem 8.1.A. These ideas work (almost) equally well for surfaces of any genus. We restrict ourselves to the genus ^ 2 case. We need a minor modification A{S) of the ideal triangulations of the Teichmiiller spaces from 5.5. The vertices of the complex A (5) are the isotopy classes (/) oiproperly embedded non-trivial (cf. 2.3) arcs / in S. The isotopies are not required to be fixed at the ends, but are required to run in the class of properly embedded arcs. A set of vertices {yo. • • •, KA? } forms a simplex of A{S) if and only if the isotopy classes yo, • " ,yn can be represented by pairwise disjoint arcs IQ, ..., In. The extended mapping class group Mod^ acts on A{S) in an obvious way. We can collapse each boundary component of S into a point (different for different boundary components) and then consider these points as punctures. Let R be the resulting surface with punctures. There is a natural map S ^^ R, establishing a diffeomorphism between the complements of the boundary and of the punctures. Clearly, this map induces an isomorphism between A(S) and the complex A(R) from 5.5. Using this isomorphism and Corollary 5.5.B, we immediately get the following result. LEMMA 8.4. A. Every automorphism ofA(S) is determined by its action on any single top dimensional simplex. In particular, if an automorphism of A(S) agrees with the action of some element of Mod^ on some simplex of codimension 0, then this automorphism agrees with this element o/Mod^ on the whole complex A(S). In view of the last lemma, the automorphisms of A (5) seem to be much more accessible than the automorphisms of C(5'). The main idea of the remaining part of the proof of Theorem 8. LA is to reduce the study of automorphisms of C{S) to a study of automorphisms of A(S). The main tool is a coding of the vertices of A(5) in terms of the vertices of C(S) and some additional data. Let us describe this coding. The vertices of A(5) are naturally divided into three types, and a vertex is coded by its type and one or two vertices of C(S), depending on its type. The types of vertices are the following: (a) the isotopy classes of arcs / connecting two different components D\, D2 of 95; (b) the isotopy classes of arcs / connecting a component D of dS with itself such that the image of this arc in R (this image is a circle passing through a puncture) does not bound a disc with one puncture; (c) the isotopy classes of arcs / connecting a component D of 95 with itself such that the image of this arc in R does bound a disc with one puncture. In addition to the type, a vertex of type (a) is coded by (C), where C is the boundary of some regular neighborhood of Di U / U D2 in 5, a vertex of type (b) is coded by the pair (Ci), (C2), where C\, C2 are two components of a regular neighborhood of D U /, and a vertex of type (c) is coded by (C), where C is the non-trivial

Mapping class groups

615

component of a regular neighborhood of D U / (one of the components is trivial in this case). Now, we need to prove that every automorphism of C(S) maps vertices coding vertices of A(S) of type (a) into similar vertices, behaves similarly with respect to types (b) and (c) and also maps codings of pairs of vertices connected by an edge in A(S) into similar codings. Figure 17 may serve as an illustration of codings of a pair of vertices connected by an edge. This is the most technically difficult part of the proof, which splits into many cases according to the types of the vertices involved. After this is done, we can assign an automorphism of A(S) to any automorphism of C(5). Moreover, one can prove that on an explicitly given codimension 0 simplex of A(S) every automorphism of A(S) induced by an automorphism of C(S) agrees with an element of Mod^, at least if the number of boundary components is > 2. In this case Lemma 8.4.A implies that the induced automorphism of A(S) is equal to some element of Mod^. And if the induced automorphism of A(S) is equal to some element of Mod^, then the original automorphism of C(S) is equal to some element of Mod^, as it is easy to see. This proves Theorem 8.1.A for surfaces with ^ 2 boundary components. The case of surfaces with ^ 1 boundary components can be reduced to the case of surfaces with ^ 2 boundary components as follows. Note first that every automorphism of C(S) takes the isotopy classes of non-separating circles into isotopy classes of non-separating circles (for example, because C is non separating if and only if there exist a vertex fi such that i(Qf, yS) = 1, where a = (C)). Since all non-separating circles are in the same orbit of the group of diffeomorphisms of S, we can assume that our automorphism of C(S) fixes some vertex a = (C), where C is non-separating. Such an automorphism induces an automorphism of the link La of a and, hence, of the complex C(Sc), where Sc is the result of cutting S along C (compare with the proof of Corollary 8.2.B). Since Sc has ^ 2 boundary components, this automorphism of C(Sc) is equal to some element of Mod^ . Considering different non-separating circles C (in fact, all of them), one can deduce that the original automorphism of C(S) is equal to some element of Mod^. (If the genus of S is 2, an additional difficulty is caused by the fact that the genus of Sc will be < 2. But our automorphisms of C(Sc) are induced by automorphisms of C(S) and, hence, retain all the nice properties of the latter, such as the preservation of the intersection number 1.) This completes our outline of the proof of Theorem 8.1 .A.

8.5. An application to subgroups of finite index As the first application of Theorem 8.1. A, we give a complete description of isomorphisms between subgroups of finite index in Mod^. This result will find its own applications in 9.2; cf. Corollary 9.2.B. 8.5.A. Suppose that S is not a sphere with ^ 4 holes and not a torus with < 2 holes. Let Fx and 7^2 be two subgroups of finite index o/Mod^. If S, in addition, is not a closed surface of genus 2, then all isomorphisms F] -^ F2 have the form x i-> gxg~^ for some g e Mod^. If S is a closed surface of genus 2, then all isomorphisms F\ -> F2 have the form x \-^ gxg~^ xi^) for some g G Mod^ and some homomorphism x from F\ THEOREM

616

N.V Ivanov

to the center o/Mod^, i.e., the subgroup o/Mod^ of order 2 generated by the hyperelliptic involution {see Theorem 7.5.D); x can be non-trivial only //T2 contains the hyperelliptic involution. PROOF. First, note that for some natural number A^ ^ 0 the A^-th powers t^ of all Dehn twists ta are contained in Fx, because Fx is of finite index. Next, Theorem 7.5.B easily implies that any isomorphism (p\F\ ^^ F2 maps sufficiently high powers of Dehn twists into powers of Dehn twists. Taking into account the fact that two non-trivial powers of Dehn twists commute if and only if the corresponding isotopy classes of circles have geometric intersection number 0 (by Theorem 7.5.C), we see that every such isomorphism (p induces an automorphism C{S) -^ C{S). By Theorem 8.1.A, this automorphism of C(5) is given by some element g G Mod^. This means that for some sufficiently large A^, we have

^Va

)-^g{a)

for some M^ / 0, for all vertices a of C{S). The potential dependence of Ma on a is irrelevant in what follows, and we write simply M for M^. Now, let f e F\. Then, for any vertex of.

On the other hand.

Comparing the results of these two computations, we conclude (note that two nontrivial powers r^, t'o of right Dehn twists are equal if and only if of = ^ and m = n) that (P(f)(g(ci)) = g(f(ct)) for all a and (after putting a = g-^P)) that (p(f){P) = gfg-^ (P) for all vertices p of C(S). If S is not a closed surface of genus 2, this implies that (p(f) = gfg~K i.e., (p has the required form; see Corollary 7.5.E. If 5 is a closed surface of genus 2, then (p(f) can differ from gfg~' by the hyperelliptic involution. It is easy to see that this difference can be described by some homomorphism x as stated. D 8.5.B. Suppose that S is not a sphere with ^ 4 holes and not a torus with ^ 2 holes. Let F be a subgroup of finite index of Mod^. Then the outer automorphism group Out(r) is finite. COROLLARY

Theorem 8.5.A was proved first by the author [95] for the case F\ = F2 = Mod^ or Mod^ and for closed surfaces only. Another, but closely related, proof in this case was provided by McCarthy [160], who also noticed additional automorphisms resulting from the hyperelliptic involution in the genus 2 case (which were overlooked in [95]). An alternative approach was suggested by Tchangang [221]. Shortly after [95] the author extended these results to surfaces with boundary and to some natural subgroups of Mod^ like F\ = F2 = PMod^; see [101]. A crucial element of these early approaches, namely the fact

Mapping class groups

617

that any isomorphism (p takes powers of Dehn twists to powers of Dehn twists, also plays a crucial role in the above proof of Theorem 8.5.A. Recently, McCarthy and the author extended the results of [101,160] to some injective homomorphisms between mapping class groups. Let us measure the size of a surface S by the number 3g — 3-\-b, where g is the genus and b is the number of boundary components of S (this number is equal to the maximal number of pairwise nonisotopic disjoint circles on S). The main result of [111,112] asserts that if the sizes of two surfaces 5, S^ differ by at most 1 and they have the genus ^ 1, then injective homomorphisms Mod^ -^ Mods' are, in fact isomorphisms and S is diffeomorphic to S^ with, possibly, a few exceptions.

8.6. An application to Teichmiiller spaces In this section we consider the Teichmiiller spaces of punctured surfaces (cf. 5.1). The Teichmiiller space TR of a punctured surface R carries a natural structure of a metric space, given by its Teichmiiller metric (we note in passing that Teichmiiller spaces have a couple of other natural metrics). This structure of a metric space is not derived from a Riemannian metric (although it can be derived from a Finslerian metric), but nevertheless has some nice geometric properties. In particular, TR is a complete metric space, every two points can be connected by a geodesic (i.e., a locally shortest path), and, moreover, geodesies have a nice description in terms of the (conformal) geometry of surfaces. For an introduction to this theory, see Abikoff's book [1]. The fact that the metric is naturally defined implies that Mod^ acts on TR by isometrics. A remarkable result proved by Royden [208] for closed surfaces (and isometrics preserving the natural complex structure of Ts), and then extended by Earl and Kra [47] to the surfaces with boundary (and general isometrics), asserts that there are no other isometrics. More precisely, we have the following theorem. 8.6. A. Suppose that R is not a sphere with ^ 4 punctures and not a torus with ^ 1 punctures. Then any isometry O/TR is induced by some element o/Mod^.

THEOREM

Note that the conclusion of this theorem is not true for a sphere with 4 punctures and a torus with 0 or 1 punctures, because for these surfaces the Teichmiiller space is isometric to the hyperbolic plane, which has a continuous group of isometrics. For a sphere with ^ 3 punctures the conclusion is vacuous, because the Teichmiiller space consists of just one point. For a torus with 2 punctures the conclusion of the theorem is not true, because if 5i,2 is a torus with 2 punctures and 5o,5 is a sphere with 5 punctures, then T^, ^ is isometric to TSQ 5, but Mod^j 2 is not isomorphic to Mod^^^ ^ (cf. 8.1). See [47]. Royden's proof (and its extension by Earl and Kra) is analytic and is based on a detailed investigation of the non-smoothness properties of the Finslerian metric underlying the Teichmiiller metric. It is essentially local in nature. A detailed exposition of this proof is presented in Gardiner's book [58]; cf. [58, Chapter 9]. Our Theorem 8.1 .A can be used to prove this theorem in a completely different manner (in all cases except that of a torus with two punctures). This new proof, outHned in [103], reveals a deep analogy between Royden's theorem and Mostow's rigidity theorems [184,185]. In fact. Theorem 8.1.A plays

618

N.VIvanov

a role in this proof similar to the role played by the theorem of Tits mentioned in 8.1 in Mostow's proof [185]. In addition to Theorem 8.1.A this proof of Theorem 8.6.A, naturally, relies heavily on the theory of the Teichmtiller spaces and, in particular, on the results of Kerckhoff [121] and Masur [152-154]. Unfortunately, this theory falls outside the scope of the present paper.

9. Mapping class groups and arithmetic groups 9.1. Arithmetic groups The analogy between arithmetic groups and mapping class groups was suggested first by Harvey [86], and had guided much of research about the mapping class groups since then. Let us give a down-to-earth version of the definition of arithmetic groups. The basic example of an arithmetic group is SL„(Z), the group of all integral n x n matrices with the determinant 1 (note that the inverse of an integral matrix with the determinant 1 is automatically an integral matrix). All other arithmetic groups can be, in some sense, constructed from this one. Suppose that G is a semisimple Q-algebraic subgroup of SL^ (R). This means that G is simultaneously a semisimple Lie subgroup of SL„ (R) and a subset of the space of all real n x n matrices which can be described by polynomial equations for the matrix entries with rational coefficients. For any such G the intersection Gz of G with the set of integral matrices is, by definition, an arithmetic group. Next, let Go be the connected component of the identity of G and (^: Go ^^ // be a homomorphism from Go onto a connected semisimple Lie group H with compact kernel (to be honest, we should also assume that H has trivial center and no compact factors, but the reader may ignore this remark). Then F = (p(Gz n Go) is, by definition, an arithmetic group, as is also any subgroup F^ of H commensurable with it, i.e., any subgroup F^ such that the intersection FHF^ has finite index both in F and F'. More precisely, we say that any such F^ is an arithmetic subgroup of H. We also call arithmetic any group isomorphic to any such F\ So, any arithmetic group can be obtained from SL„ (Z) in three steps: taking an intersection with a semisimple Q-algebraic real Lie group, then taking the image under a surjective homomorphism with compact kernel (this step is the most difficult to control), and, finally, taking any subgroup commensurable with this image. In particular, any subgroup of finite index of an arithmetic group is also an arithmetic group. In addition to SL„ (Z), a typical example of an arithmetic group is the symplectic group Sp2o(Z), which shows up in the topology of surfaces as the group of all automorphisms of the first homology group ^ i (5^) of a closed orientable surface Sg of genus g preserving the intersection pairing. In particular, we have a natural homomorphism Mod^^, -^ Sp2^(Z), which is well known to be surjective. In [86], Harvey asked if the mapping class groups are arithmetic. They turned out to not be, as was proved by the author [95,100]. On the other hand, the years after [86] profoundly demonstrated that there is indeed a deep analogy between mapping class groups and arithmetic groups, especially in their cohomology properties (the main results of Section 6 are, in fact, motivated by analogous results about arithmetic groups). Although the first proof

Mapping class groups

619

of the non-arithmeticity of the mapping class groups was quite short (cf. 9.3), it did not exhaust the problem, and the question of why the mapping class groups are not arithmetic led to further results. In the next section (cf. 9.2) we will describe what seems to be the best answer to this question.

9.2. Abstract commensurators The notion of the abstract commensurator of a group has its roots in the theory of arithmetic groups. The author learned it from [190]; see also [5, Appendix B], where it is attributed to Serre and Neumann. It allows us to express one of the most striking differences between arithmetic and non-arithmetic groups; cf. Theorem 9.2.A below and the remarks after it. Let r be a group. Let us consider all possible isomorphisms cp'.Fx ^^ r2 between subgroups Fi, Fi of finite index of F. Let us identify two such isomorphisms (p, (p' defined on F\, F[ respectively, if they agree on a subgroup of finite index in the intersection F\ n F[. We can compose them in an obvious manner; the composition (p o (p' of (p:F\ ^^ F2 with (p': F[ -^ F^ is defined on (^~^ (7^2 Pi F[). Under this composition operation, the classes of such isomorphisms form a group, which is called the abstract commensurator of F and is denoted by Comm(r). There is a natural map i\F -^ Comm(r), sending an element y of F to the (class of the) inner automorphism g i-> ygy~^. This map is injective if the centralizers of subgroups of finite index in F are trivial, as is the case for arithmetic groups and for mapping class groups (see Theorem 7.5.D for the latter). As a nice example, let us mention that Comm(Z") = SL„ (Q). THEOREM

9.2. A. If F is an arithmetic group, then i (F) is of infinite index in Comm(r).

PROOF. We only outline the main idea. Let us consider SL„ (Z) and show that any element of SL^ (Q) naturally defines an element of Comm(SL„ (Z)). Let g e SLn (Q), and let m be the least common multiple of all denominators of the entries of the matrices g,g~^. Let F\ ={y e SL„ (Z) | y = / modm^}, where / is the identity matrix. Clearly, F\ is of finite index in SL„ (Z). If y e F\, then y = I -\- m^h, where h is an integral matrix, and hence gyg~^ = I -\- m^ghg~^ is also an integral matrix. Thus gF\g~^ C SL„ (Z). Let us show that F2 = gF\g~^ is a subgroup of finite index in SL„ (Z). Consider the subgroup F^ = {y e SL„ (Z) \ y = I modm"^}. If y G F\ then y = I -\- m'^h, where h is an integral matrix, and hence g~^yg = I -\- m^(m^g~^hg) e F\, because m^g~^hg is an integral matrix. It follows that g~^F'g C F\, and hence F' c gF\g~^. Since F' obviously has finite index in SL„(Z), we conclude that F2 = gF\g~^ D F' is of finite index. Now, it is clear that the inner automorphism x i-> gxg~^ induces an isomorphism F\ -^ F2 between two subgroups of finite index of SL„(Z), and hence an element of Comm(SL„(Z)). This leads to a homomorphism SL„(Q) -^ Comm(SL„(Z)). Clearly, its image contains /(SL^(Z)) as a subgroup of infinite index. This proves the theorem for SL,(Z). The same argument works for the groups Gz, where G is a semisimple Q-algebraic group, simply because Gz, hke SL„(Z), has a corresponding group of rational matrices

620

N.VIvanov

associated to it. After this, it is not hard to extend the result to the groups of the form (^(Gz n Go) (cf. 9.1) and then to any group commensurable with such a group. We refer to Zimmer's book [246] for the details; cf. [246, Section 6.2]. D A converse to this theorem is also true: if F is a lattice in a semisimple Lie group G (i.e., if G/r has finite invariant volume) and ((F) is of infinite index in Comm(r), then F is arithmetic. This result is an immediate corollary of an arithmeticity theorem of Margulis (cf. [246, Chapter 6]) and Mostow's rigidity theorem [184,185]; cf. [190, Section 1]. This converse is much more deep and difficult than Theorem 9.2.A, and we do not need it for our applications. 9.2.B. Suppose that S is not a sphere with ^ 4 holes and not a torus with ^ 2 holes. Then the mapping class groups Mod^, Mod^, and all subgroups of finite index in them, are not arithmetic.

COROLLARY

It follows from Theorem 8.5.A that Comm(Mod5) = Comm(Modp = /(Modp, and hence Mod^, Mod^ cannot be arithmetic by Theorem 9.2.A. If r is a subgroup of finite index in Mod^, then again Comm(r) = /(Modp by Theorem 8.5.A. (Note that the additional automorphisms for the closed surfaces of genus 2 disappear when we pass to a subgroup of finite index not containing the hyperelliptic involution.) D PROOF.

If 5 is a sphere with ^ 4 holes or a torus with ^ 1 holes, then Mod^ is indeed arithmetic: if 5 is a torus with ^ 1 holes, then Mod^ is isomorphic to SL2 (Z); if 5 is a sphere with 4 holes, then Mod^ is commensurable with PSL2(Z) = SL2(Z)/{±/} (cf., [101], Section 7 for a detailed discussion of this case); if 5 is a sphere with ^ 3 holes, then Mod^ is finite. If 5 is a torus with two holes, then Mod^ is not arithmetic. The above argument does not (currently) apply to this case, because Theorem 8.1.A does not include it. But older approaches, discussed in the next two sections (cf. 9.3 and 9.4) do apply to this case also. It is worth stressing that this proof of the non-arithmeticity of the mapping class groups uses only very basic properties of arithmetic groups, going not much further than the definition, and a deep result (Theorem 8.1.A) from the topology of surfaces. This contrasts strongly with the original approach to the non-arithmeticity, which relied on the deepest properties of arithmetic groups; cf. 9.3.

9.3. The original approach to the non-arithmeticity Arithmetic groups, Hke representations, can be divided into reducible and irreducible ones. An arithmetic group F is called reducible if it is "close" to a product of two infinite groups; in particular, if F is reducible, then it contains two subgroups F\, F2 of infinite index commuting with each other and generating together a subgroup of finite index in F. Using the last remark and Theorem 7.5.A, it is easy to see that no subgroup of finite index in Mod^ can be isomorphic to a reducible arithmetic group. If F is not reducible, it is called irreducible.

Mapping class groups

621

By the definition, any arithmetic group F arises as a subgroup of a real semisimple Lie group H (without compact factors). The rank of the latter (i.e., the dimension of a Cartan subalgebra of its Lie algebra) is usually called the rank of T. In this section we will not address the question of whether this rank depends only on F or also on the realization of F as an arithmetic subgroup of some semisimple Lie group (cf. 9.4), and will consider our use of this notion as a slight abuse of language. The main point is that the properties of F depend strongly on whether the rank of T is 1 or ^ 2. In particular, if F is an irreducible arithmetic group of rank ^ 2, then any normal subgroup of F is either a central finite subgroup or a subgroup of finite index. This is the famous Margulis finiteness theorem; see [246, Chapter 8] for the details. Now, for a closed surface Sg of genus g the kernel of the natural homomorphism Mod^^, -> Sp2^ (Z) from 9.1 is an infinite (and non-central) subgroup of infinite index. Indeed, this kernel, usually called the Torelli subgroup of Mod^^, contains, for example, all Dehn twists about separating circles and the above homomorphism is surjective, as we noted in 9.1. It follows that Mod^^ cannot be an irreducible arithmetic group of rank ^ 2. Similar arguments apply to surfaces with non-empty boundary; for example, one can use the kernels of homomorphisms from 2.8 instead of the Torelli subgroup or a version of the latter. It remains to show that Mods cannot be an irreducible arithmetic group of rank 1. One way to show this is to use the well known fact that any such arithmetic group contains a subgroup of finite index (actually, any torsion-free subgroup of finite index will work) which is isomorphic to the fundamental group of a complete Riemannian manifold of finite volume with curvature pinched between two negative constants, together with the following theorem. 9.3.A. If S is not a sphere with < 4 holes and not a torus with ^ 1 holes, then no subgroup of finite index in Mod^ is isomorphic to the fundamental group of a complete Riemannian manifold of finite volume with curvature pinched between two negative constants. THEOREM

PROOF. We only indicate the main idea. It is sufficient to prove that any subgroup of finite index in Mod^ contains a subgroup isomorphic to (Z * Z) x Z, and that none of the considered fundamental groups does. Subgroups of Mod^^ isomorphic to (Z * Z) x Z can be constructed in abundance with powers of Dehn twists as generators. The absence of such subgroups in the considered fundamental groups follows easily from the theory of Eberlein and O'Neill [48]; see [100], proof of Theorem 2 for the details. D

Irreducible arithmetic groups of rank 1 can also be excluded in a couple of other ways. See [79], Section 4.2 for a cohomological approach due to Harer, and still another approach due to Goldman. The latter is similar in spirit to the use of the Eberlein-O'Neill theory above. The ideas outlined in this section, if combined with Mostow's rigidity theorem [184,185], lead also to some non-arithmeticity results for normal subgroups of Mod^ (even for so-called subnormal subgroups). For example, the Torelli subgroup, like the mapping class group itself, is not arithmetic. See [100].

622

N. V. Ivanov

Finishing the discussion of this original approach to the non-arithmeticity, we note that it uses almost no non-trivial information about mapping class groups, but relies heavily on the Margulis finiteness theorem, one of the deepest results about arithmetic groups at all. Cf. the remarks at the end of 9.2.

9.4. Rank of the mapping class groups As we mentioned in 9.3, properties of an arithmetic group depend to a considerable extent on whether its rank is 1 or > 2. Given the analogy between mapping class groups and arithmetic groups, it is important to ascertain what arithmetic groups - of rank 1 or of rank ^ 2 - would be better analogues of mapping class groups. The best answer to this question would be a notion of rank applicable to both arithmetic groups and mapping class groups, together with a computation of this rank for the mapping class groups. A suitable notion of rank was proposed by Ballmann and Eberlein [4], who developed the ideas of Prasad and Ragunathan [206]. Let F be an abstract group. Let (F)/ denote the set of elements of F whose centralizer contains a free Abelian subgroup of rank ^ / as a subgroup of finite index (so that (r)o C iF)\ C • • • C (F)/ C • • •)• Let r{F) be the least natural number / such that F can be presented as a finite union F =

yi(F)iU'--Uy^(F)i

of left translates of the set {F)i (the elements y i , . . . , y^ G /" are arbitrary). Let us define the rank of F as the maximum of the numbers r{F^), where F^ runs over all subgroups of finite index in F. We denote it by rank(F). Prasad and Raghunathan [206] showed that if F is an arithmetic group, then r(F) is equal to its rank in the sense of 9.3. In particular, this implies that the rank of an arithmetic group F in the sense of 9.3 depends only on the structure of F as an abstract group and not on its particular realization as an arithmetic subgroup of a semisimple Lie group. The replacement of r (•) by rank() was suggested by Ballmann and Eberlein [4] in order to get a notion of rank invariant under passing to subgroups of finite index. They proved that if r is a discrete group of motions of a complete simply connected Riemannian manifold X whose curvature is nonpositive and bounded from below, such that X/F has finite volume, then rank(r) is equal to a geometrically defined rank of X. This result, in fact, includes the case of arithmetic groups, and means that this rank of an abstract group gives the desired result even for groups more general than arithmetic ones. THEOREM

9.4. A. IfSis not a sphere with < 3 holes, then rank(Mod5) = 1.

PROOF. We will give only an outline of the proof. Let F be a torsion free subgroup of finite index in Mod^ (such subgroups exist in view of Corollary 7.1.B). Since rank() is invariant under passing to subgroups of finite index, it is sufficient to show that r(F) = 1 for any such T. It is easy to see that r(F) ^ 0. Let TpA denote the set of pseudo-Anosov elements of F. Theorem 7.5.A easily implies that the centralizers of pseudo-Anosov ele-

Mapping class groups

623

ments contain an infinite cyclic subgroup as a subgroup of finite index. Thus FpA C (P) \. Hence it suffices to prove that for some finite number of elements g i , . . . , g^ € F we have /^ = gi^pAU--'Ug^rpA. In other words, it is sufficient to find a finite number of elements g i , . . . , gm such that for any g e F one of the elements g j~ ^ g, • • •, g^ ^ g is pseudo-Anosov. Using the compactness of the Thurston boundary of the Teichmiiller space (cf. 7.2), it is possible to show that there exists a finite collection of isotopy classes a i , . . . , a„ such that for any isotopy class fi one of the pairs of/, ^ fills S in the sense of 7.1. Fix such a collection Qfi,..., Qf„. Let ^z, 1 < / ^ «, be a non-trivial power of the Dehn twist taj contained in F (remember that F is of finite index in Mod^). Using Fathi's Theorem 7.1.D, one can show that for any g e F one of the elements tfg, 1 ^ / ^n, 1 ^ J ^ 8, is pseudo-Anosov. It follows that r ( r ) = 1. We refer to [102] for the details. D COROLLARY 9.4.B. For any compact orientable surface S, the group Mods is not isomorphic to any arithmetic group of rank ^ 2.

One can replace all references to the Margulis finiteness theorem in 9.3 by this corollary. This leads to still another proof of the non-arithmeticity of the mapping class groups. On the arithmetic groups side, it uses the results of Prasad and Raghunathan [206], which are not as difficult as the Margulis finiteness theorem. It also uses some quite non-trivial results on the mapping class groups side (Fathi's Theorem 7.1.D); cf. the remarks at the end of 9.2, 9.3. The next theorem, as we will explain in a moment, also supports the idea that the mapping class groups are similar to the arithmetic groups of rank 1, as opposed to rank ^ 2. THEOREM 9.4.C. Suppose that S is not a sphere with ^ 3 holes. Let us fix a finite set of generators of Mod^ and let us denote by \ - \y/ the minimal word length with respect to these generators. If f is an element of infinite order in Mod^, then there is a constant c > 0 such that

\r\w^c\n\ for all n. It is easy to see that one can always replace / by a nontrivial power of / in the proof. In particular, we may assume that / is a pure element of Mod^ (see 7.1). Then there are two different cases to consider. If the diffeomorphism induced on the result of cutting S along some realization of a{f) is isotopic to a pseudo-Anosov diffeomorphism on at least one component, then the theorem follows from the results of Mosher [182]. Otherwise, / is a composition of Dehn twists about several disjoint circles. This case was recently dealt with by Minsky [166], in response to a question posed by the author. Recall that an arithmetic group F arises as a subgroup of a real semisimple Lie group H. The group F is called cocompact if H/F is compact. The similarity of Mod^ with cocompact arithmetic groups can be easily excluded by cohomological reasons (as, for example.

624

N.VIvanov

in [79, Section 4.2]; cf. 9.3). But, for noncompact arithmetic groups F of rank ^ 2, there are always elements f e F of infinite order such that the word length \f^\w grows slower than linearly (in fact, logarithmically), according to a recent result of Lubotzky, Mozes and Raghunathan [144]. Therefore, Theorem 9.4.C supports the analogy between Mod^ and rank 1 arithmetic groups. See [166] for other results of Farb, Lubotzky and Minsky supporting this analogy.

9.5. The Mostow-Margulis superrigidity The Marguhs superrigidity theorem [149], (cf. also [150] or [246, Chapter 5]) extends the Mostow rigidity theorem [184,185]. It asserts, in particular, that any homomorphism Fx -> F2 between arithmetic subgroups F\, F2 of semisimple Lie groups H\, H2 respectively (without compact factors) extends to a homomorphism H\ -^ H2 after, perhaps, passing to a subgroup of finite index in F\, if rankTi ^ 2. Note that the restriction rank F\ ^2in this theorem is necessary. Here we meet one of the situations where arithmetic groups of rank 1 and rank > 2 behave differently. The analogy between mapping class groups and arithmetic groups suggests that one may extend this theorem to include mapping class groups on the equal footing with arithmetic groups. At first sight, this is impossible, because there are no Lie groups naturally (or, to the best of our knowledge, otherwise) containing mapping class groups. But this simply means that there should be no homomorphisms, or better, taking into account the possible passage to a finite index subgroup in the Margulis theorem, that any homomorphism should have finite image. In addition, since mapping class groups have rank 1 and the source group in the Margulis theorem should have rank ^ 2, we are forced to consider only homomorphisms from arithmetic groups of rank > 2 to mapping class groups. Then it is natural to expect that any such homomorphism has finite image. The first step in the direction of this conjecture was made in [95], and the conjecture itself was stated explicitly in [97]. Further steps were made in [100]. Recently, Kaimanovich and Masur [118] almost completely proved it. Following [118], let us call a subgroup of Mod^ non-elementary if it leaves no finite subset of the Thurston boundary of the corresponding Teichmtiller space invariant. THEOREM 9.5. A. Any non-elementary subgroup of a mapping class group is not isomorphic to an arithmetic group 6>/rank ^ 2.

See [118], Theorem 2.4.1. The proof of Kaimanovich and Masur is based on a theory of random walks on Teichmtiller spaces and mapping class groups developed by Masur [155] and Kaimanovich and Masur [118] and the results of Furstenberg [56,57] about random walks and harmonic functions on discrete subgroups of Lie groups ([57] provides an excellent introduction to this circle of ideas). More precisely, Furstenberg introduced a property of harmonic functions (which are closely related to the random walks) on discrete groups which holds for all arithmetic groups of rank ^ 2 and does not hold for (at least some) arithmetic groups of rank 1. Kaimanovich and Masur show that this property does not hold also for all non-elementary subgroups of mapping class groups; thus they cannot

Mapping class groups

625

be arithmetic groups of rank ^ 2. Note that these results of Furstenberg (as also the results of Mostow) preceded and influenced Margulis' approach to his superrigidity theorem. If combined with the Margulis finiteness theorem and the results of [108] (cf. 7.4) about subgroups of mapping class groups, Theorem 9.5.A leads to the following corollary, which proves the above conjecture. COROLLARY 9.5.B. Let cp: F ^^ Mods be a homomorphismfrom an arithmetic group F to a mapping class group Mod^. If rankF ^ 2, then the image ofcp is finite.

A proof of this corollary along similar lines was also found by Farb and Masur [51]. It seems that it is possible to prove this corollary more directly, avoiding the use of the Margulis finiteness theorem, of random walks (and, hence, of Furstenberg's results), and even of Teichmiiller spaces (but one still needs their Thurston's boundaries). Such an approach would follow the Margulis proof of his superrigidity theorem. I hope to return to these ideas in a future paper.

Acknowledgement I am very grateful to R. Sher for the careful reading of this paper.

References [1] W. Abikoff, The Real Analytic Theory of Teichmiiller Space, Lecture Notes in Math., Vol. 820, Springer (1980). [2] D.V. Anosov, Geodesic flows on compact Riemannian manifolds of negative curvature, Proc. Steklov Math. Inst. 90 (1967); EngUsh transl. Amer. Math. Soc. (1969). [3] R. Baer, Isotopien auf Kurven von orientierbaren, geschlossenen Fldchen, J. Reine Angew. Math. 159 (1928), 101-116. [4] W. Ballmann and R Eberlein, Fundamental groups of manifolds of nonpositive curvature, J. Differential Geom. 25(1) (1987), 1-22. [5] H. Bass and R. Kulkami, Uniform tree lattices, J. Amer. Math. Soc. 3 (4) (1990), 843-902. [6] H. Bass and A. Lubotzky, Automorphisms of groups and of schemes of finite type, Israel J. Math. 44 (1) (1983), 1-22. [7] M. Bauer, Examples of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc. 330 (1) (1992), 333370. [8] H. Behr, Uber die endliche Definierbarkeit verallgemeinerter Einheitgruppen, J. Reine Angew. Math. 211 (1962), 123-135. [9] D.J. Benson, Specht modules and the cohomology of mapping class groups, Mem. Amer. Math. Soc. 90 (443) (1991), 29-92. [10] D.J. Benson and F.R. Cohen, The mod 2 cohomology of the mapping class group for a surface of genus two, Mem. Amer. Math. Soc. 90 (443) (1991), 93-104. [11] D.J. Benson and P.H. KrophoUer, Cohomology of groups. Handbook of Algebraic Topology, I.M. James, ed., Elsevier (1995), 917-950. [12] P. Bergau and J. Mennicke, Uber topologishe Abbildungen der Bretzekflache vom Geschlecht 2, Math, Z. 74 (1960), 414^35. [13] R. Bieri and B. Eckmann, Groups with homological duality generalizing the Poincare duality. Invent. Math. 20 (2) (1973), 103-124.

626

N. V. Ivanov

[14] J.S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure App. Math. 22 (1969), 213-238. [15] J.S. Birman, Abelian quotients of the mapping class group of a 2-manifold, Bull. Amer. Math. Soc. 76 (1) (1970), 147-150. Errata: Bull. Amer. Math. Soc. 77 (3) (1971), 479. [16] J.S. Birman, Braids, Links and Mapping Class Groups (based on lecture notes by James Cannon), Ann. of Math. Stud., Vol. 82, Princeton University Press (1974). [17] J.S. Birman, The algebraic structure of surface mapping class groups. Discrete Groups and Automorphic Functions, W.J. Harvey, ed.. Academic Press (1977), 163-198. [18] J.S. Birman, Mapping class groups of surfaces. Braids (Santa Cruz, 1986), J.S. Birman and A. Libgober, eds, Contemp. Math. 78 (1988), 13-43. [19] J.S. Birman and H.M. Hilden, On mapping class groups of closed surfaces as covering spaces. Advances in the Theory of Riemann Surfaces, L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra and H.E. Ranch, eds, Ann. Math. Stud., Vol. 66, Princeton University Press (1971), 81-115. [20] J.S. Birman, A. Lubotzky and J. McCarthy, Abelian and solvable subgroups of the mapping class group, Duke Math. J. 50 (4) (1983), 1107-1120. [21] J.S. Birman and B. Wajnryb, Errata: presentations of the mapping class group, Israel J. Math. 88 (1-3) (1994), 425^27. [22] C.-F Bodigheimer, On the topology of moduli spaces of Riemann surfaces. Part II: Homology operations. Preprint, Mathematica Gottingensis, Heft 23 (1990). [23] A. Borel and J.-P. Serre, Corners and arithmetic groups. Comment. Math. Helv. 48 (4) (1973), 436^91. [24] B.H. Bowditch and D.B.A. Epstein, Natural triangulations associated to a surface. Topology 27 (1) (1988), 91-117. [25] G.E. Bredon, Geometry and Topology, Graduate Texts in Math., Vol. 139, Springer (1993). [26] K.S. Brown, Cohomology of Groups, Graduate Texts in Math., Vol. 87, Springer (1982). [27] K.S. Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1) (1984), 1-10. [28] K.S. Brown, Buildings, Springer (1989). [29] P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Math., Vol. 106, Birkhauser (1992). [30] A.J. Casson and S.A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, Vol. 9, Cambridge Univ. Press (1989). [31] J. Cerf, Topologie de certaines espaces de plongements. Bull. Soc. Math. France 89 (1961), 227-380. [32] J. Cerf, Sur les diffeomorphisms de la sphere de dimension trois (7~4 = 0), Lecture Notes in Math., Vol. 53, Springer (1968). [33] J. Cerf, La stratification naturelle des espaces de functions differentiables relies et le theoreme de la pseudo-isotopie, Publ. Math. IHES 39 (1970), 5-173. [34] R. Chamey and F Cohen, A stable splitting for the mapping class group, Michigan Math. J. 35 (2) (1988), 269-284. [35] R. Chamey and R. Lee, Characteristic classes for the classifying spaces of Hodge structures, /T-Theory 1 (3) (1987), 237-270. [36] R. Chamey and R. Lee, An application ofhomotopy theory to mapping class groups, J. Pure Appl. Algebra 44 (1-3) (1987), 127-135. [37] F.R. Cohen, Homology of the mapping class groups for surfaces of low genus. The Lefschetz Centennial Conference, Part II. Proceedings on Algebraic Topology, S. Gitler, ed., Contemp. Math., Vol. 58 (1987), 21-30. [38] F.R. Cohen, Artin's braid group and the homology of certain subgroups of mapping class groups, Mem. Amer. Math. Soc. 90 (443) (1991), 6-28. [39] F.R. Cohen, On the mapping class groups for punctured spheres, the hyperelliptic mapping class groups, S0(3), and Spin^(3), Amer. J. Math. 115 (2) (1993), 389^34. [40] F.R. Cohen, Mapping class groups and classical homotopy theory. Mapping Class Groups and Moduli Spaces of Riemann Surfaces, C.-F. Bodigheimer and R. Hain, eds, Contemp. Math., Vol. 150 (1993), 63-74.

Mapping class groups

627

[41] F.R. Cohen, On mapping class groups from a homotopy theoretic point of view. Algebraic Topology: New Trends in Localization and Periodicity, C. Broto, C. Casacuberta and G. Mislin, eds. Progress in Math., Vol. 136, Birkhauser (1996), 119-142. [42] F.R. Cohen and U. Tillmann, Toward homology operations for mapping class groups, Homotopy Theory via Algebraic Geometry and Group Representations (Evanston, XL, 1997), M. Mahowald and S. Priddy, eds, Contemp. Math., Vol. 220 (1998), 35-46. [43] M. Dehn, Papers in Group Theory and Topology, Translated and Introduced by John Stillwell, Springer (1987). [44] M. Dehn, On curve systems on two-sided surfaces, with application to the mapping problem. Lecture (supplemented) to the math, colloquium, Breslau 11-2-1922, In: [43], 234-252. [45] M. Dehn, Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 135-206; EngUsh transl.: [43], 256362. [46] C.J. Earle and J. Eells, A fibre bundle description ofTeichmiiller theory, J. Differential Geom. 3(1) (1969), 19^3. [47] C.J. Earle and I. Kra, On isometrics between Teichmiiller spaces, Duke Math. J. 41 (3) (1974), 583-591. [48] P Eberlein and B. O'Neill, Visibility manifolds. Pacific J. Math. 46 (1) (1973), 45-109. [49] D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson and W.P. Thurston, Word Processing in Groups, Jones and Barlett PubUshers (1992). [50] D.B.A. Epstein and R.C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1) (1988), 67-80. [51] B. Farb and H. Masur, Superrigidity and mapping class groups. Topology 37 (6) (1998), 1169-1176. [52] A. Fathi, Dehn twists and pseudo-Anosov diffeomorphisms. Invent. Math. 87 (1) (1987), 129-151. [53] A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Seminaire Orsay, Asterisque, Vol. 66-67, Soc. Math, de France (1979). [54] W. Fenchel, Bemaerkinger om endlige grupper af afbildungklasser. Mat. Tidsskr. B (1950), 90-95. [55] R. Fricke and F. Klein, Vorlesungen Uber die Theorie der automorphen Funktionen. I, Die gruppentheoretischen Grundlagen, Teubner, Leipzig (1897). [56] H. Furstenberg, Poisson boundaries and envelopes of discrete groups. Bull. Amer. Math. Soc. 73 (3) (1967), 350-356. [57] H. Furstenberg, Random walks and discrete subgroups of Lie groups. Advances in Probability and Related Topics, Vol. 1, Marcel Dekker (1971), 3-63. [58] F P Gardiner, Teichmiiller Theory and Quadratic Differentials, Wiley (1987). [59] S. Gervais, Presentation and central extension of mapping class groups. Trans. Amer. Math. Soc. 348 (8) (1996), 3097-3132. [60] S. Gervais, A finite presentation of the mapping class group of an oriented surface. Preprint (1998), 22 pp. [61] J. Gilman, Structures of elliptic irreducible subgroups of the modular group, Proc. London. Math. Soc. 47 (1) (1983), 2 7 ^ 2 . [62] H. Glover and G. Mislin, On the stable cohomology of the mapping class groups. Algebraic Topology, Gottingen 1984, L. Smith, ed.. Lecture Notes in Math., Vol. 1172 (1985), 80-84. [63] H. Glover and G. MisUn, Torsion in the mapping class group and its cohomology, J. Pure Appl. Algebra 44 (1-3) (1987), 177-189. [64] H.H. Glover, G. Mislin and Y. Xia, On the Parrel cohomology of mapping class groups. Invent. Math. 105 (3) (1992), 535-545. [65] H.H. Glover, G. Mislin and Y. Xia, On the Yagita invariant of mapping class groups. Topology 33 (3) (1994), 557-574. [66] L. Greenberg, Maximal groups and Riemann surfaces. Discontinuous Groups and Riemann Surfaces, L. Greenberg, ed., Ann. of Math. Stud., Vol. 79, Princeton University Press (1974), 207-226. [67] M. Gromov, Hyperbolic groups. Essays in Group Theory, S.M. Gersten, ed., MSRI Publications, Vol. 8, Springer (1987), 75-263. [68] E. Grossman, On the residually finiteness of certain mapping class groups, J. London Math. Soc. 9 (1) (1974), 160-164. [69] R.M. Hain, Completions of mapping class groups and the cycle C — C~, Mapping Class Groups and Moduli Spaces of Riemann Surfaces, C.-F. Bodigheimer and R. Hain, eds, Contemp. Math., Vol. 150 (1993), 75-105.

628

N.VIvanov

[70] R.M. Hain, Torelli groups and geometry of moduli spaces of curves. Current Topics in Algebraic Geometry (Berkeley, CA, 1992/93), MSRI Publications, Vol. 28, Cambridge University Press (1995), 97-143. [71] R.M. Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (3) (1997), 597-651. [72] R.M. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves. Algebraic Geometry Santa Cruz 1995, Proc. Sympos. Pure Math., Vol. 62, Part 2, Amer. Math. Soc. (1997), 97-142. [73] M.-E. Hamstrom, The space of homeomorphisms of a torus, Illinois J. Math. 9 (1965), 59-65. [74] M.-E. Hamstrom, Homotopy groups of the space of homeomorphisms of a 2-manifold, Illinois J. Math. 10 (1966), 563-573. [75] J.L. Harer, The second homology group of the mapping class group of an orientable surface. Invent. Math. 72 (2) (1983), 221-239. [76] J.L. Harer, The homology of the mapping class group and its connection to surface bundles over surfaces, Four-Manifold Theory, C. Gordon and R. Kirby, eds, Contemp. Math., Vol. 35 (1984), 311-314. [77] J.L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (2) (1985), 215-249. [78] J.L. Harer, The virtual cohomology dimension of the mapping class group of an orientable surface. Invent. Math. 84 (1) (1986), 157-176. [79] J.L. Harer, The cohomology of the moduli space of curves. Lecture Notes in Math., Vol. 1337, Springer (1988), 138-221. [80] J.L. Harer, Stability of the homology of the moduli spaces ofRiemann surfaces with spin structure. Math. Ann. 287 (2) (1990), 323-334. [81] J.L. Harer, The third homology group of the moduli space of curves, Duke Math. J. 63 (1) (1991), 25-55. [82] J.L. Harer, The rational Picard group of the moduli space ofRiemann surfaces with spin structure. Mapping Class Groups and Moduli Spaces of Riemann Surfaces, C.-F. Bodigheimer and R. Hain, eds. Contemp. Math., Vol. 150 (1993), 107-136. [83] J.L. Harer, The fourth homology group of the moduli space of curves. Preprint (1993), 36 pp. [84] J.L. Harer, Improved homology stability for the homology of the mapping class groups of surfaces. Preprint (1993), 18 pp. [85] J.L. Harer and D. Zagier, The Euler characteristic of the moduli space of curves. Invent. Math. 85 (3) (1986), 457-485. [86] W.J. Harvey, Geometric structure of surface mapping-class groups, Homological Group Theory, C.T.C. Wall, ed., London Math. Soc. Lecture Notes Series, Vol. 36, Cambridge University Press (1979), 255-269. [87] W.J. Harvey, Boundary structure of the modular group, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, I. Kra and B. Maskit, eds, Ann. of Math. Stud., Vol. 97, Princeton University Press (1981), 245-251. [88] A. Hatcher, Homeomorphisms of sufficiently large P^-irreducible 3-manifolds. Topology 15 (4) (1976), 343-347. [89] A. Hatcher, On triangulations of surfaces. Topology Appl. 40 (2) (1991), 189-194. [90] A. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint. Topology Appl. 30 (1) (1998), 63-88. [91] A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface. Topology 19 (3) (1980), 221-237. [92] J. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 2>-manifolds. Acta Math. 142 (1-2) (1979), 123-155. [93] S.P. Humphries, Generators for the mapping class group. Topology of Low-Dimensional Manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., Vol. 722 (1979), 44-47. [94] N.V Ivanov, Groups of diffeomorphisms ofWaldhausen manifolds. Studies in Topology II, Notes of LOMI Scientific Seminars, Vol. 66 (1976), 172-176; English transl.: J. Soviet Math. 12 (1) (1979), 115-118. [95] N.V Ivanov, Algebraic properties of the Teichmiiller modular group, Dokl. Akad. Nauk SSSR 275 (4) (1984), 786-789; Enghsh transl.: Soviet Math. Dokl. 29 (2) (1984), 288-291. [96] N.V. Ivanov, On the virtual cohomology dimension of the Teichmiiller modular group. Lecture Notes in Math., Vol. 1060, Springer (1984), 306-318. [97] N.V Ivanov, Algebraic properties of mapping class groups of surfaces, Banach Center Publications, Vol. 18, PWN - Pohsh Scientific PubUshers (1986), 15-35.

Mapping class groups

629

[98] N.V. Ivanov, Complexes of curves and the Teichmiiller modular group, Uspekhi Mat. Nauk 42 (3) (1987), 49-91; English transl.: Russian Math. Surveys 42 (3) (1987), 55-107. [99] N.V. Ivanov, Subgroups of Teichmiiller modular groups and their Frattini subgroups, Funktsional. Anal, i Prilozhen. 21 (2) (1987), 76-77; Enghsh transl.: Functional Anal. Appl. 21 (2) (1987), 154-155. [100] N.V. Ivanov, Teichmiiller modular groups and arithmetic groups. Research in Topology 6, Notes of LOMI Scientific Seminars, Vol. 167 (1988), 95-110; Enghsh transl.: J. Soviet Math. 52 (1) (1990), 2809-2818. [101] N.V. Ivanov, Automorphisms of Teichmiiller modular groups. Lecture Notes in Math., Vol. 1346, Springer (1988), 199-270. [102] N.V. Ivanov, Rank of Teichmiiller modular groups. Mat. Zametki 44 (5) (1988), 636-644; Enghsh transl.: Math. Notes 44 (5-6) (1988), 829-832. [103] N.V. Ivanov, Automorphisms of complexes of curves and of Teichmiiller spaces. Preprint IHES/M/89/60, 1989, 13 pp.; Also in: Progress in Knot Theory and Related Topics, Travaux en Cours, Vol. 56, Hermann, Paris (1997), 113-120. [104] N.V. Ivanov, Stabilization of the homology of Teichmiiller modular groups. Algebra i Analiz 1 (3) (1989), 110-126; English transl.: Leningrad J. of Math. 1 (3) (1990), 675-691. [105] N.V. Ivanov, Attaching corners to Teichmiiller space. Algebra i Analiz 1 (5) (1989), 115-143. English transl.: Leningrad J. of Math. 1 (5) (1990), 1177-1205. [106] N.V. Ivanov, Complexes of curves and Teichmiiller spaces. Mat. Zametki 49 (5) (1991), 54—61; English transl.: Math. Notes 49 (5-6) (1991), 479^84. [107] N.V. Ivanov, On the finite approximability of modular Teichmiiller groups, Sibirsk. Mat. Zh. 32 (1) (1991), 182-185; Enghsh transl.: Siberian Math. J. 32 (1) (1991), 148-150. [108] N.V. Ivanov, Subgroups of Teichmiiller Modular Groups, Transl. Math. Monographs, Vol. 115, Amer. Math. Soc. (1992). [109] N.V. Ivanov, On the homology stability for Teichmiiller modular groups: Closed surfaces and twisted coefficients. Mapping Class Groups and Moduh Spaces of Riemann Surfaces, C.-F. Bodigheimer and R. Hain, eds, Contemp. Math., Vol. 150 (1993), 149-194. [110] N.V. Ivanov, Automorphisms of complexes of curves and of Teichmiiller spaces, Intemat. Math. Res. Notices 14 (1997), 651-666. [ I l l ] N.V. Ivanov and J.D. McCarthy, On injective homomorphisms between Teichmiiller modular groups. Preprint (1995), 85 pp. [112] N.V. Ivanov and J.D. McCarthy, On injective homomorphisms between Teichmiiller modular groups I, Invent. Math., 135 (2) (1999), 425-486. [113] D.L. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1)(1979), 119-125. [114] D.L. Johnson, A survey of the Torelli group, Low-Dimensional Topology (San Francisco, Calif., 1981), Contemp. Math., Vol. 20 (1983), 165-179. [115] D.L. Johnson, The structure of the Torelli group. I. A finite set of generators for X, Ann. of Math. 118 (3) (1983), Al'i-Ml. [116] D.L. Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves. Topology 24 (2) (1985), 113-126. [117] D.L. Johnson, The structure of the Torelli group. III. The abelianizaton ofT, Topology 24 (2) (1985), 127-144. [118] V.A. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group. Invent. Math. 125 (2) (1996), 221-264. [119] W. van der Kallen, Homology stability for linear groups. Invent. Math. 60 (3) (1980), 269-295. [120] M. Kapovich, Hyperbolic Manifolds and Discrete Groups, Progress in Math, Vol. 183, Birkhauser, 2001. [121] S.P Kerckhoff, The asymptotic geometry of Teichmiiller spaces. Topology 19 (1) (1980), 2 3 ^ 1 . [122] S.P Kerckhoff, The Nielsen realization problem. Bull. Amer. Math. Soc. 2 (3) (1980), 452^54. [123] S.P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (2) (1983), 235-265. [124] S.P. Kerckhoff, The geometry of Teichmiiller space, Proc. of the International Congress of Mathematicians, Vols. 1, 2 (Warsaw, 1983), PWN - PoUsh Scientific Pubhshers, Warsaw (1984), 665-678. [125] S.P Kerckhoff, Lines of minima in Teichmuller space, Duke Math. J. 65 (2) (1992), 187-213. [126] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1) (1992), 1-23.

630

N.VIvanov

[127] M. Korkmaz, Complexes of curves and mapping class groups, PhD Thesis, Michigan State University (1996); Also in: M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95 (2) (1999), 85-111. [128] A. Kosinslcy, Differential Manifolds, Pure and Appl. Math., Vol. 138, Academic Press (1993). [129] J.L. Koszul, Lectures on groups of transformations, Notes by R.R. Simha and R. Sridharan, Tata Institute of Fundamental Research, Bombay (1965). [130] F. Laudenbach, Presentation du groupe de diffeotopies d'une surface compacte orientable. In: [53], Exp. 15, 267-282. [131] R. Lee and S. Weintraub, Cohomology of Sp{4, Z) and the related groups and spaces. Topology 24 (4) (1985), 391^10. [132] W.B.R. Liclcorish, A representation of orientable combinatorial 3-manifolds, Ann. of. Math. 76 (3) (1962), 531-540. [133] W.B.R. Liclcorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Phil. Soc. 59 (2) (1963), 307-317. [134] W.B.R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Phil. Soc. 60 (4) (1964), 769-778. [135] W.B.R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Phil. Soc. 61 (1965), 61-64. [136] W.B.R. Lickorish, A foliation of 3-manifolds, Ann. of. Math. 82 (3) (1965), 414-420. [137] W.B.R. Lickorish, Corrigendum: On the homeotopy group of a 2-manifold, Proc. Cambridge Phil. Soc. 62 (4) (1966), 679-681. [138] D.D. Long, Constructing pseudo-Anosov maps. Knot Theory and Manifolds (Vancouver, BC, 1983), Lecture Notes in Math., Vol. 1144 (1985), 108-114. [139] D.D. Long, A note on the normal subgroups of mapping class groups. Math. Proc. Cambridge Philos. Soc. 99(1) (1986), 79-87. [140] D.D. Long and H. Morton, Hyperbolic 3-manifolds and surface homeomorphisms. Topology 25 (4) (1986), 575-583. [141] E. Looijenga, Intersection theory on Deligne-Mumford compactifications {affter Witten and Kontsevich], Seminaire Bourbaki, Vol. 1992/1993, Exp. 768, Asterisque, Vol. 216 (1993), 187-212. [142] E. Looijenga, Cellular decompositions of compactified moduli spaces ofpointed curves. The Moduli Space of Curves, Progress in Math., Vol. 129, Birkhauser (1995), 369-^00. [143] E. Looijenga, Stable homology of the mapping class group with symplectic coefficients and the universal Abel-Jacobi map, J. Algebraic Geom. 5 (1) (1996), 135-150. [144] A. Lubotzky, S. Mozes and M.S. Raghunathan, Cyclic subgroups of exponential growth and metric on discrete groups, C. R. Acad. Sci. Paris 317 (8) (1993), 735-740. [145] F. Luo, On nonseparating simple closed curves in a compact surface. Topology 36 (1997), 381^10. [146] F. Luo, A presentation of the mapping class group. Math. Res. Lett. 4 (5) (1997), 735-739. [147] F. Luo, Automorphisms of complexes of curves. Topology 39 (2) (2000), 283-298. [148] W. Magnus, Uber Automorphismen von Fundamentalgruppen berandeter Fldchen, Math. Ann. 109 (1934), 617-648. [149] G.A. Margulis, Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math. 76 (1) (1984), 93-120. [150] G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Math., 3 Folge, B. 17, Springer (1991). [151] G.A. Margulis and G.A. Soifer, Maximal subgroups of infinite index in finitely generated linear groups, J. Algebra 69(1) (1981), 1-23. [152] H. Masur, On a class of geodesies in Teichmiiller space, Ann of Math. 102 (2) (1975), 205-221. [153] H. Masur, Uniquely ergodic quadratic differentials. Comment. Math. Helv. 55 (2) (1980), 255-266. [154] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. 115 (1) (1982), 169-201. [155] H. Masur, Random walks on Teichmiiller spaces and the mapping class group, J. Anal. Math. 67 (1995), 117-164. [156] H. Masur and Y. Minsky, Geometry of the complexes of curves I: Hyperbolicity, Invent. Math. 138 (1) (1999), 103-149.

Mapping class groups

631

[157] H. Masur and Y. Minsky, Geometry of the complexes of curves I: Hierarchical structure, Geom. Funct. Anal. 10 (4) (2000), 902-91 A. [158] J.D. McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes. Preprint (1982). [159] J.D. McCarthy, A "Tits-alternative" for subgroups of surface mapping class groups. Trans. Amer. Math. Soc. 291 (2) (1985), 583-612. [160] J.D. McCarthy, Automorphisms of surface mapping class groups. A recent theorem ofN. Ivanov, Invent. Math. 84(1) (1986), 49-71. [161] J. McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975), 205-213. [162] C.T. McMuUen, Renormalization and 3-Manifolds which Fiber over the Circle, Ann. Math. Stud., Vol. 142, Princeton University Press (1996). [163] G. Mess, Unit tangent bundle subgroups of the mapping class groups. Preprint IHES/M/90/30 (1990), 15. [164] E.Y. Miller, The homology of the mapping class group, J. Differential Geom. 24 (1) (1986), 1-14. [165] Y.N. Minslcy, A geometric approach to the complex of curves on a surface. Topology and Teichmiiller Spaces (Katinkulta, 1995), World Sci. (1996), 149-159. [166] Y.N. Minslcy, Dehn twists have linear growth. Preprint (1997), 5 pp.; Also in: B. Farb, A. Lubotzlcy and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke Math. J. 106 (3) (2001), 581-597. [167] G. Mislin, Mapping class groups, characteristic classes, and Bernoulli numbers. The Hilton Symposium 1993. Topics in Topology and Group Theory, G. Mishn, ed., CRM Proceedings & Lecture Notes, Vol. 6, Amer. Math. Soc. (1994), 103-131. [168] Sh. Morita, Characteristic classes of surface bundles. Bull. Amer. Math. Soc. 11 (2) (1984), 386-388. [169] Sh. Morita, Characteristic classes of surface bundles. Invent. Math. 90 (3) (1987), 552-577. [170] Sh. Morita, Characteristic classes of surface bundles and bounded cohomology, A Fete of Topology, Academic Press (1988), 233-257. [171] Sh. Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. I, Ann. Inst. Fourier, Grenoble 39 (3) (1989), 777-810. [172] Sh. Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. II, Math. Proc. Cambridge Phil. Soc. 105 (1) (1989), 79-101. [173] Sh. Morita, Casson's invariant for homology ?>-spheres and characteristic classes of surface bundles. I, Topology 28 (3) (1989), 305-323. [174] Sh. Morita, Mapping class groups of surfaces and three-dimensional manifolds. Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Vols. I, II, Math. Soc. Japan (1991), 665-674. [175] Sh. Morita, Characteristic classes of surface bundles and Casson invariant, Sugaku 43 (3) (1991), 232247; Enghsh transl.: Sugaku Expositions 7 (1) (1994), 59-79. [176] Sh. Morita, The structure of the mapping class group and characteristic classes of surface bundles. Mapping Class Groups and Moduli Spaces of Riemann Surfaces, C.-F. Bodigheimer and R. Hain, eds. Contemp. Math., Vol. 150 (1993), 303-315. [177] Sh. Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group. Invent. Math. I l l (1) (1993), 197-224. [178] Sh. Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (3) (1993), 699-726. [179] Sh. Morita, Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles, J. Differential Geom. 47 (3) (1997), 560-599. [180] L. Mosher, The classification of pseudo-Anosovs, Low Dimensional Topology and Kleinian Groups (Coventry/Durham, 1984), London Math. Soc. Lecture Notes Ser., Vol. 112, Cambridge Univ. Press (1986), 13-75. [181] L. Mosher, Mapping class groups are automatic. Math. Res. Lett. 1 (2) (1994), 249-255. [182] L. Mosher, Mapping class groups are automatic, Ann. of Math. 142 (2) (1995), 303-384. [183] L. Mosher, A user's guide to the mapping class group: once punctured surfaces. Geometric and Computational Perspectives on Infinite Groups (Minneapohs, MN and New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 25 (1996), 101-174. [184] G.D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Publ. Math. IHES 34 (1968), 53-104.

632

N. V. Ivanov

[185] G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Stud., Vol. 78, Princeton University Press (1973). [186] D. Mumford, Abelian quotients of the Teichmuller modular group, J. Anal. Math. 18 (1967), 227-244. [187] D. Mumford, A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1) (1971), 289-294. [188] D. Mumford, Towards an enumerative geometry of the moduli space of curves. Arithmetic and Geometry, Vol. II, Progress in Math., Vol. 36, Birkhauser (1983), 271-328. [189] M. Naatanen and R.C. Penner, The convex hull construction for compact surfaces and the Dirichlet polygon. Bull. London Math. Soc. 23 (6) (1991), 568-574. [190] W.D. Neumann and A.W. Reid, Arithmetic of hyperbolic manifolds. Topology '90, B. Apanasov, W.D. Neumann, A.W. Reid and L. Siebenmann, eds, Walter de Gruyter (1992), 273-310. [191] J. Nielsen, Untersuchungen zur Theorie der geschlossenen zweiseitigen Fldchen I, Acta Math. 50 (1927), 189-358. [192] J. Nielsen, Untersuchungen zur Theorie der geschlossenen zweiseitigen Fldchen II, Acta Math. 53 (1929), 1-76. [193] J. Nielsen, Untersuchungen zur Theorie der geschlossenen zweiseitigen Fldchen III, Acta Math. 58 (1932), 87-167. [194] J. Nielsen, Abbildungsklassen der endlicher Ordnung, Acta Math. 75 (1942), 23-115. [195] J.-P. Otal, Le theoreme d'hyperbolisation pour les varietes fibress de dimension 3, Asterisque, Vol. 235, Soc. Math, de France (1996). [196] R. Palais, Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34 (1960), 305312. [197] R.C. Penner, The decorated Teichmiiller space ofpunctured surfaces, Comm. Math. Phys. 113 (2) (1987), 299-339. [198] R.C. Penner, The moduli space of a punctured surface and perturbative series. Bull. Amer. Math. Soc. 15 (1) (1986), 73-77. [199] R.C. Penner, Perturbative series and the moduli space ofRiemann surfaces, J. Differential Geom. 27 (1) (1988), 35-53. [200] R.C. Penner, A construction ofpseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc. 310 (1) (1988), 179-197. [201] R.C. Penner, Weil-Petersson volumes, J. Differential Geom. 35 (3) (1991), 559-608. [202] R.C. Penner, Universal constructions in Teichmuller theory. Adv. in Math. 98 (2) (1993), 143-215. [203] V P Platonov, Frattini subgroups of linear groups andfinitary approximability, Dokl. Alcad. Naulc SSSR 171 (1966), 798-801; English transl.: Soviet Math. Dokl. 7 (6) (1966), 1557-1560. [204] T. Poston and I.N. Stewart, Taylor expansions and catastrophes. Pitman Res. Notes Math., Vol. 7 (1976). [205] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (3) (1978), 347-350. [206] G. Prasad and M.S. Raghunathan, Cartan subgroups and lattices in semisimple groups, Ann. of Math. 96 (2) (1972), 296-317. [207] S. Priddy and Y. Xia, Cohomological invariants of mapping class groups. Quart. J. Math., Oxford 47 (187) (1996), 361-370. [208] H.L. Royden, Automorphisms and isometrics of Teichmuller spaces. Advances in the Theory of Riemann Surfaces, L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra, H.E. Ranch, eds, Ann. of Math. Stud., Vol. 66, Princeton University Press (1971), 369-383. [209] P. Schmutz, Die Parametrisierung des Teichmiillerraumes durch geoddtische Ldngenfunktionen, Comment. Math. Helv. 68 (2) (1993), 278-288. [210] G.P Scott, The space of homeomorphisms of a 2-manifold, Topology 9 (1970), 97-109. [211] H. Seifert, Bemerkungen zur stetigen Abbildungen von Fldchen, Abh. Math. Sem. Univ. Hamburg 12 (1937), 23-37. [212] J.-P. Serre, Rigiditi de foncteur d'Jacobi d'echelon n ^ 3, Sem. H. Cartan (1960/1961), Appendix to Exp. 17. [213] J.-P. Serre, Cohomologie des groupes discretes. Prospects in Mathematics, Ann. of Math. Stud., Vol. 70, Princeton University Press (1971), 77-169. [214] J.-P. Serre, Arbres, amalgames, SL2, redige avec la collaboration de Hyman Bass, Asterisque, Vol. 46 (1977); EngUsh transl.: Trees, translated by J. Stillwell, Springer (1980).

Mapping class groups [215] [216] [217] [218] [219] [220]

[221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241]

[242] [243] [244] [245] [246]

633

S. Smale, Dijfeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959), 621-626. E. Spanier, Algebraic Topology, 2nd ed.. Springer (1990). J. Stillwell, Geometry of Surfaces, Universitexts, Springer (1992). K. Strebel, On quadratic differentials with closed trajectories and second order poles, J. Anal. Math. 19 (1967), 373-382. K. Strebel, Quadratic Differentials, Ergebnisse der Math., 3 Folge, Band 5, Springer (1984). D. Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et sur les varietes hyperboliques de dimension 3fibrees sur S\ Seminaire Bourbald, Vol. 1979/80, Exp. 554, Lecture Notes in Math., Vol. 842, Springer (1981), 196-214. R.T. Tchangang, Le groupe d'automorphismes du groupe modulaire, Ann. Inst. Fourier (Grenoble) 37 (2) (1987), 19-31. W.P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (2) (1988), 417-431. W.P. Thurston, Hyperbolic structures on 3-manifolds IT. Surface groups and 3-nuinifolds which fiber over the circles. Preprint (1986). U. Tillmann, Discrete models for the category ofRiemann surfaces. Math. Proc. Cambridge Phil. Soc. 121 (1) (1997), 39-^9. U. Tillmann, On the homotopy of the stable mapping class group. Invent. Math. 130 (1997), 257-275. J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer (1974). A.J. Tromba, Teichmiiller Theory in Riemannian Geometry (based on notes by J. Dezler), Lectures in Math. ETH Zurich, Birkhauser (1992). A.J. Tromba, Dirichlet's energy on Teichmiiller moduli space and the Nielsen realization problem. Math. Z. 222 (3) (1996), 451^64. B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (2-3) (1983), 157-174. B. Wajnryb, Mapping class group of a surface is generated by two elements. Topology 35 (2) (1996), 377-383. B. Wajnryb, An elementary approach to the mapping class group of a surface, Geom. Topology 3 (1999), 405^66 (electronic). E. Witten, Two dimensional gravity and intersection theory on moduli spaces. Surveys in Differential Geometry, Vol. 1 (1991), 243-310. S.A. Wolpert, Geodesic length functionals and the Nielsen problem, J. Differential Geom. 25 (1987), 275295. Y. Xia, The p-period of an infinite group, Publ. Mat. 36 (1) (1992), 241-250. Y. Xia, The p-periodicity of the mapping class group and an estimate of its p-period, Proc. Amer. Math. Soc. 116 (4) (1992), 1161-1169. Y Xia, The p-torsion of the Farrel-Tate cohomology of the mapping class group /^(p_i)/2, Topology '90, B. Apanasov, WD. Neumann, A.W Reid and L. Siebenmann, eds, Walter de Gruyter (1992), 391-398. Y Xia, The p-torsion of the Farrell-Tate cohomology of the mapping class group rp__\, J. Pure Appl. Algebra 78 (3) (1992), 319-334. Y Xia, An obstruction in the mapping class group. Bull. London Math. Soc. 27 (1) (1995), 51-57. Y Xia, On the cohomology of Fp, Trans. Amer. Math. Soc. 347 (9) (1995), 3659-3670. Y Xia, On the connectivity of posets in the mapping class groups. Algebraic Topology: New Trends in Localization and Periodicity, C. Broto, C. Casacuberta and G. Mislin, eds, Progress in Math., Vol. 136, Birkhauser (1996), 393^00. Y Xia, Geodesic length functions and cohomology of Fp, Preprint (1995), 14 pp. D. Zagier, On the distribution of the number of cycles of elements in symmetric groups, Nieuw Archief voor Wiskunde 13 (3) (1995), 489-495. H. Zieschang, Discrete groups of motions of the plane and plane group diagrams, Uspekhi Mat. Nauk 29 (3) (1966), 195-212 (in Russian). H. Zieschang, E. Vogt and H.-D. Coldeway, Surfaces and Planar Discontinuous Groups, Lecture Notes in Math., Vol. 835, Springer (1980). R.J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs Math., Vol. 81, Birkhauser (1984).

This Page Intentionally Left Blank

CHAPTER 13

Seifert Manifolds Kyung Bai Lee University of Oklahoma, Norman, OK 73019, USA

Frank Raymond University of Michigan, Ann Arbor, MI 48109, USA

Contents 1. Introduction 2. Definitions, motivation and examples 2.1. Rough definition of Seifert fiber spaces 2.2. Torus actions 2.3. Examples of injective actions 2.4. Injective Seifert

fibering

2.5. T O P G ( G XW)

637 637 637 639 641 641 642

2.6. Example: 3-dimensional Seifert manifolds with base the 2-torus 2.7. Injective Seifert constructions 2.8. The singular fibers 2.9. Infra-homogeneous spaces 3. Group cohomology 3.1. Introduction 3.2. Group extensions 3.3. Fullback and pushout 3.4. Extensions with Abelian kernel A and H^(Q; A) 3.5. Extensions with non- Abelian kernel G and H^iQ; Z(G)) 3.6. H\Q; C) with a non-Abelian C 3.7. Mapping one group extension into another 4. Seifert fiber space construction with TOP(7 (G x W) 4.1. Introduction 4.2. Mapping an extension of G into T O P G (G X W) 4.3. Construction with T O P G ( G X W) 4.4. The meaning of existence, uniqueness and rigidity 4.5. Proofs of Theorem 4.3.2 4.6. Smooth case 5. Applications HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

635

645 647 648 649 650 650 651 653 654 655 656 656 661 661 662 665 666 669 672 673

636

K.B. Lee and F. Raymond

5.1. Introduction 5.2. Existence of closed smooth K{n, l)-manifolds 5.3. When is 9 injective? 5.4. Rigidity of infra-homogeneous spaces 5.5. Lifting problem for homotopy classes of self-homotopy equivalences 5.6. Seifert fiberings with codimension 2 fibers 5.7. The classical Seifert 3-dimensional manifolds 6. Reduction of the universal group 6.1. Purpose and requirements 6.2. Injective holomorphic Seifert fiberings 6.3. PSL(2,]R)-geometry • • .^ 6.4. Lorentz structures and PSL(2, E)-geometry 6.5. Deformation spaces for PSL(2, M)-geometry 6.6. Polynomial structures on virtually poly-Z manifolds References

673 673 676 676 679 682 684 685 685 685 688 693 695 701 703

Seifert manifolds

637

1. Introduction Seifert fibered spaces have played an interesting and important role in topology and transformation groups. The classical definition of Seifert [60] in 1933, concerning circle bundles with specified singularities over 2-dimensional manifolds, preceded much of the notions and development of bundle theory. In fact, Seifert's splendid paper had a profound influence on both bundle theory and low-dimensional topology. Recognition of the importance of Seifert fiber spaces to transformation groups was communicated a long time ago to one of the authors by Deane Montgomery. Many of the interesting spaces in topology and geometry admit the structure of a bundle with singularities. The geometric structure is usually displayed as part of this bundle structure. While Seifert fiberings, in their utmost generality, could be said to encompass all of these phenomena, we have concentrated here, in our exposition, on a particular type called injective Seifert fiberings. These seem to entail the simplest structure in that they admit a global uniformization, have close connections with transformation groups, and have many applications. Moreover, because topologists have become intrigued with the "geometrization" of manifolds, we hope that this paper will also inform the reader of the significant role that Seifert fiber spaces and Seifert constructions can play in these matters. We have tried to make our exposition as concrete and accessible as possible. This is especially true in the beginning and in the last two sections where examples and illustrations of the theory are worked out in detail. Each section has its own introduction and it will be helpful to read Sections 2, 5 and 6 first. To digest Sections 3 and 4, it is advisable to read these through first for an overview before checking the details. Notation. We shall use the following notation throughout the rest of the paper. For a topological group G with K a closed subgroup and X a nice topological space, we put Aut(G) Inn(G) Out(G) NG{K) CG(K) Aut(G, K) lnn(G,K) Out(G, K) ix{a) TOP(X) Diff(X)

= = = = = = = = = = =

Continuous automorphisms of G Inner automorphisms of G Aut(G)/Inn(G) Normalizer of/^ in G Centrahzer ofKinG {ae Aut(G): a\K e Aui(K)} Inn(G) H Aut(G, i^) Aut(G, i^)/Inn(G, K) Conjugation by a\ so, /x(a)(jc) = axa~^ for JC G G The group of self-homeomorphisms of X The group of self-diffeomorphisms of a smooth manifold X

2. Definitions, motivation and examples 2.1. Rough definition of Seifert fiber spaces In Section 2, we explain what we mean by Seifert fiber spaces, give motivations, precise definitions and some examples.

638

K.B. Lee and F. Raymond

The Seifert fiber space construction was first defined and studied in [12] and [13] and later reformulated in [34,38] and [41]. It is really a generalization of the classical Seifert spaces and homogeneous spaces of a Lie group by its uniform lattices. We shall start with a rough definition, continue with some motivation, and finally, give an exact definition. We then examine what one must do to effectively construct all possible injective Seifert fiberings over a given base with a given typical fiber. As a result of this investigation, we determine the precise nature of the singular fibers. A Seifert fibering F -^ E ^^ B is a. "fibering" with singularities. For each b e B, there is associated a finite subgroup Qb of TOP(F), the group of self-homeomorphisms of F, so that p~^ (b) = F/Qt,. If Qb is trivial, p~\b) = F is called a typical fiber. Otherwise F/Qb is called a singular fiber. For example, if F = T^ is a /:-torus, and Qb acts freely on F, then F/Qb will be a flat manifold which is finitely covered by the torus. We shall require the groups Qb to be controlled in a particular way as a replacement of the local triviality condition. Before giving a formal definition, we start with some examples. EXAMPLE

2.1.1. As a simple example, consider the Mobius band

£ = /x[-l,l]/(0,0^(1,-0. It has an obvious circle action so that its orbit space B = S^\E is an interval. Thus, we get S^ -> E -> B. The orbits are circles parallel to the boundary circle. The center circle orbit however is doubly covered by each of the other orbits. B is an arc [0,1]. Here p~\t) = S^, a free orbit for every t ^ 0, while p ' ^ O ) = Z2 \ 5 ^ the center circle, is the only singular fiber. EXAMPLE 2.1.2. Let a : r^ ^ r^ be the map Qf(zi, zi) = (z\, zi) on a 2-torus. Then a has period 2, and has 4 fixed points. In fact, B = T^/Z2 is a sphere with 4 branch points. Take the product T^ x I,a 2-torus and the unit interval. Identify the two ends by

to form the mapping torus M of a. Denote an element of M by [w,t], and define an 5^-action on M by

where the second coordinate is taken modulo 1 and compatible with the identifications. The space E = (T^ x /)/Z2 is homeomorphic to an orientable 3-dimensional flat Riemannian manifold with holonomy group Z2. The circle S^ acts as isometrics on E.lf p denotes the orbit mapping, then the orbit space is B = T^/Z2. The free orbits, away from the 4 branch points, are typical fibers while the doubly covered orbits over the branch branch points are singular fibers. These singular fibers are still circles, but are half of the length of the typical fibers.

Seifert manifolds

639

2.2. Torus actions To facilitate our understanding of Seifert fiber spaces we shall examine first the situation arising from group actions. We shall assume familiarity with elementary properties of group actions. Good references as we shall use these concepts are the book by Bredon [7], the paper [12], and the chapter of this Handbook by Adem and Davis [1]. We recall first several definitions and facts. Our spaces X are path-connected, completely regular and Hausdorff so that the various slice theorems are valid. For covering space theory, we need and tacitly assume that our spaces are locally path-connected and semi 1-connected. A topological group G acts properly on X if the closure of {^ G G | gC fl C 7^ 0} is a compact subset of G, for all compact subsets C of X. See [49] for properties of proper actions. In particular, a proper action of a (not necessarily connected) locally compact Lie group on a completely regular space admits a slice, and the orbit space is Hausdorff. If 2 is a discrete group acting properly on X, we shall say that Q acts properly discontinuously on X. Note the isotropy subgroups are finite for a properly discontinuous action. Let G be a path-connected group acting on a path-connected Hausdorff space X. Fix X eX. The evaluation map ev^ : (G, e) -^ (X, jc) is defined by

This induces the evaluation homomorphism ev^:7ri(G,^)^7ri(X,x). Put H = image of evj c TTI (X, JC). Then / / is a central subgroup of TTI (X, JC) which is independent of choices of JC [12, Lemma 4.2]. Let ^ be a normal subgroup of 7t\ (X, jc) and XK the covering space of X associated with the subgroup K so that 7t\ {XK ,X^) = K. Assume that H C K and put Q = 7ti (X, jc)/;ri (XK , x) = itx (X, x)/K. Then we have the PROPOSITION 2.2.1 ([12, §4] and [7, Chapter I, §9]). (1) The G-action on X lifts to an action O/XK which commutes with the covering Q-action on XK- The lifted action ofG on XK is proper if the action ofGonX is proper (2) The induced Q action on the quotient space W = G\ XK is properly discontinuous (i.e., behaves at least like branched coverings with Hausdorff quotient) provided G acts properly on X. DEFINITION 2.2.2. A torus action (T^, X) is said to be injective if ev^ :7ri(r^,^) -^ 7T\ (X, jc) is injective.

640

K.B. Lee and F. Raymond

Suppose T^ acts injectively on a path connected, locally path connected, semi 1-connected paracompact Hausdorff space X. Then, by the above proposition, the T^ action Hfts to an action on X^k, where Z^ is the image of 7TI(T^) ^^ 7T\ (M). We show that the lifted action (T^, Xjk) is free and splits. PROPOSITION 2.2.3 ([12, §3.1]). IfT^ acts injectively on X, then X^k splits into T^ xW so that {T^, Xj^k) = (T^, T^ x W), where the T^ action on T^ x W is via translation on thefirstfactor and trivial on the simply connected W factor

PROOF. Let G = T^ and 5^ be a subgroup of G isomorphic to the circle. (Note that S^ is a direct factor.) Lift the S^ action to X' = X^^k. Let y^ e X^ and suppose 5^ 7^ 1 is the stabilizer of the lifted S^ action ai y\ Choose paths y in X^ from the base point x^ over X to y^ and a : (/, 0, 1) -> (5*, 1, z) where z is the "first" element 7^ 1 of 5^ for which z-y^ = y\ Then a(t)y^ defines a loop at j ' ; and y * a * y is the associated loop based at x^ Now ( } / * a * y ) " ~ ) / * Q f " * } 7 represents a generator of Z^ = / / . Hence n is the order of z in 5 ^ This implies that}/ * a * y represents an nth root of a generator of Z^ which is impossible unless n = l. Thus SL = 1, for all y^ G X^ and all circle subgroups of 5^ of 7^. Hence the lifted toral action (7^, X') must be a free action and X -> W = T^ \ X' is a. principal T^ bundle. We now show that this bundle is trivial. Let f :W ^^ Bjk be the classifying map from W into the classifying space for principal T^-bundles. The bundle X' ^^ W is represented by the homotopy class [/] € [W, Bjk^. Since Bjk is a Kij}, 2) space, [/] is represented by the image / * ( M ) , where u € H^(Bjk; Z^) represents id: Bjk -^ Bjk. The cohomology sequence,

0 -^

H^W; Z^) -^ H\r-1})

-^ H\T^\ I}) -^ H^{W\ Z^)

—> //^(X^Z^), which arises from terms of low degree of the spectral sequence of the fibering X^ ^^ W, is exact (e.g., [43, p. 332, Ex. 2], or [11, p. 329 C]). We observe that H^{X'\l!^) -^ H^{T^\7}) is a surjective isomorphism because TC\ (T^, 1) -^ 7t\ (X\ jcO is an isomorphism. Consequently, /*(M) injects into H^{X'\ l}). On the other hand, 7r*(/*(M)) = f*(n^*(u)), where TT'\Ejk -^ Bjk, the universal T^b u n d l e , a n d / i s t h e b u n d l e m a p Z ' ^ £7^^ inducedby/. But7r'*(w) G ^^(^7^^; Z^) = 0. Therefore, 7r*(/*(w)) = 0, which implies /*(M) = 0. Hence, X' -> W is a trivial fibering and X' = T^ X W. Finally, we remark that the group T^ xQis acting properly on T^ x W, and W is simply connected. The T^ action does not Uft to the universal covering X of X but the induced ineffective R^ action onT^ xW lifts to an effective R^ action on R^ x W and commutes with the group TTI (X, x) of covering transformations on R^ x W. D

Seifert manifolds

641

2.3. Examples ofinjective actions A manifold M" is called hyper-aspherical [15] if there is a degree 1-map from M to a closed aspherical manifold of dimension n. PROPOSITION 2.3.1. Any effective torus action on the following closed connected manifolds is injective: (1) All aspherical manifolds [12]. (2) All hyper-aspherical manifolds [15]. (3) More generally, all manifolds M"" with ^*: H''(K(7ti(M), 1);Q) -^ //"(M;Q) surjective, where §* is induced by the classifying map § : M ^- K(7r\ (M), 1), with 7t{(M) torsion free. (The torsion freeness assumption of 7T\{M) can be dropped if we assume the action is smooth, cf [8] and [40].) See [8,15,20,28,59,62] and [40] for such generalizations. (4) Homologically Kdhler manifolds all of whose isotropy subgroups are finite {e.g., holomorphic actions) [6, p. 170], [13, p. 186]. If every circle action on a closed manifold M is injective, the only compact connected Lie groups that can act on M are tori. See [12] for a proof of this fact. Therefore tori are the only compact connected Lie groups that act effectively on those manifolds Usted in (1), (2) and (3). Moreover, in each of these cases, n\ (T^) must inject into the center of Tt\ (M) if T^ acts effectively. Consequently, the only compact Lie groups that can act effectively on these M for which the center of 7t\ (M) is finite are finite groups.

2.4. Injective Seifert fib ering Perhaps the most important feature of an injective torus action (r^, X) is the spUtting (r^, T^ X W) in Proposition 2.2.3. The universal covering group M^ acts ineffectively via R^ ^ r^ on r^ X W and lifts to the action of # as left translations on R^ x W, the universal covering space of T^ x W and of X. This R^ action contains the l} covering transformations over T^ x W and commutes with the group n =7Z\ (X) of covering transformations on X. Together R^ and n generate a subgroup R^ • TT of TOP(R^ x W) isomorphic to R^ x^/t TT, since Z^ = R^ Pi TT. We have been describing these features of injective actions to motivate our definition of injective Seifert fiberings. It is apparent that we could start with R^ x W and using the reverse procedure, reconstruct the injective toral actions on X. We will now describe this reverse procedure. We will also put it in a more general context where R^ is replaced by any Lie group G and W is any completely regular space admitting covering space theory. Let TT be a discrete subgroup of TOP(G x W) and assume: 1. The left translational action t{G) of G on G x W is normalized by jx. 2. r CTt (~\ 1(G) is discrete and normal in n. 3. The induced action of Q = 7t/r onW is proper. These conditions imply that the group 1(G) • n c TOP(G x W), generated by G and TT, acts properly on G x W.

642

K.B. Lee and F. Raymond

With the above three conditions, we obtain the commutative diagram: G\

G r\ r\G

^ G XW ^\ —^

X = IT\{G^W)

^ W

Product principal G-fibering

Q\

- ^

Q\W

Seifert fibering

X is called an injective Seifert fiber space, with typical fiber F \G, B is called the base and the mapping p is called the injective Seifert fibering. Since the actions of TT on G x W and the action of Q on W are properly discontinuous, the quotient spaces X and B are reasonable spaces. If n centralizes 1(G), then Fen ni(G) is in the center of G and the G action descends to a G / r action on X. In general, neither F is central in G nor n centralizes the G action but, as we shall see, the obvious "fibers" (inverse images p~\b)) still have a very nice description.

2.5. T O P G ( G X W)

To describe or construct all possible injective Seifert fiberings over B with typical fiber r \ G, we begin with a proper action of a discrete group Q on W so that B = Q\W. We form the product space GxW letting G act on the first factor of G x W by left translations. Denote this action by 1{G) C TOP(G x W). We then search for discrete group n contained in the normalizer of 1(G) in TOP(G x W) such that T c TT fi G is discrete in G and the induced action of the quotient n/F is equivalent to the Q action on W. Our first task then is to describe the algebraic structure of the normalizer of 1(G) in TOP(G x W). The topological structure of TOP(G x W), while important and significant for some applications, is not necessary for our problem now at hand. A space is admissible if it is completely regular (Hausdorff) locally path-connected and semi-locally simply-connected. Let G be a Lie group, Aut(G) the Lie group of continuous automorphisms of G. TOP(X) denotes the group of homeomorphisms of X. See Section 1 for a list of notation. Recall that a homeomorphism / of G x W onto itself is weakly G-equivariant if and only if there exists a continuous automorphism a/ of G so that f(a'X,w)

=

for all a € G and (jc, w) £G DEFINITION

af(a)f(x,w) xW.

2.5.1. The group of all weakly G-equivariant self-homeomorphisms of G x

W is denoted by T O P G ( G X W). LEMMA

2.5.2.

PROOF.

Let / e TOP(G x W). Then the following are all equivalent:

T O P G ( G X W)

is the normalizer of i(G) in TOP(G x W),

Seifert manifolds (1) / G T O P G ( G X

643

W).

(2) / is weakly G-equivariant. (3) There exists Uf e Aut(G) such that f{a - x,w) = af(a)f(x, w) for all a G G and (x,w)eG X W. (4) There exists of/ e Aut(G) such that / o i(a) o / " ^ = l(afia)) for all aeG. (5) / normalizes €(G). ° Each element / G T O P G ( G X W ) sends fibers of GxW induces a map / G T O P ( W ) SO that

^W

to fibers. That is, each /

G XW - ^ Gx W

1

W

^—^ W

commutes. Therefore, to each / G T O P G ( G Aut(G) X TOP(W). This assignment T O P G ( G XW)—^

X W),

we can assign a pair ( a / , / ) G

Aut(G) X TOP(W)

is a surjective homomorphism and it splits. For, if we define (a, h){x, w) = (a(x), h(w)) for (Of, h) G Aut(G) x TOP(W), then (a,h){a • x,w) = (a(a - x),h(w)) =

{a(a)(a(x),h(w))

=

l{oi{a)){a(x),h(w))

=

i{a(a))(a,h)(x,w).

Let K be the kernel of T O P G ( G X W) ^ Aut(G) x TOP(W). If f eK, since / moves only along the fibers, there exists a unique continuous function A,: G x W ^^ G such that / ( x , w) = (x ' (A(x, w))~\ w). We show that k only depends upon W. For any a e G, ax ' (^X(ax, w)~\ w) = f(a - x, w) = 1(a) f(x, w) =

since a/ = id

t(a){x'{X(x,wy\w))

= ax ' (X{x, w)~\ If). So, X(ax, w) = X(x, w) which means X only depends upon W. Let M(W, G) = {the continuous maps of W into G}.

644

K.B. Lee and F. Raymond

For each A G M ( W , G ) , define fx^Khy fx(x, w) = (x - (X(w))~\ w). The assignment M(W, G) B X\-^ fx e K is easily checked to be an isomorphism, where the group law in M(W, G) is (Ai •A-2)(M;)=AI(M;) •A2(W;).

Conjugation in T O P G ( G is given by

X W)

induces the action of Aut(G) x TOP(W) on M(W, G). It

that is, (a,h) oXo (a,h) \x,w) = (x-a[x[h

\w)))

-1

,w),

for (of,/z) o A, o (of,/i) ^(jc, If) = (Qf,/z) o A(a \x),h = =

\w))

(a,h){a-\x)'{k{h-\w))y\h-\w)) {x-a{X{h-\w)))~\w)

and ( ^' X)(x,w) = a =

oXoh~\x,w) {x'a{x{h-\w)))~\w).

Therefore we have shown LEMMA 2.5.3. T O P G ( G XW)= malizer of i{G) m TOP(G x W).

M(W, G)

>^ (Aut(G) x TOP(W)), and it is the nor-

The group operation in T O P G ( G X W) is

(Ai,ai,/zi)-(A2,a2,/22) = (Ai •^"''^'^X2,ai 00^2,/^i 0/12) = (AI • (ofi o A,2 o/z~ ),Qfi ooi2,h\ 0/12). Specifically, (A,, a, /z) acts on (x, u;) by (A,Qf,/z) • (x, w;) = ((A,, 1,1) o (l,Qf,/z))(x, w;) = (A,l,l)(a(x),/i(w;)) =

{a{x)^{x{h{w)))-\h{w)).

Seifert manifolds

645

Then M(W, G) xi Aut(G) is the group of all weakly G-equivariant homeomorphisms of G X.W which move only along the fibers. The group M(W, G) is the gauge group for the trivial fiber bundle G y.W ^^W. For a G G, the constant map W ^^ G sending W to a is denoted hy r{a). Clearly, r{a) = (a, 1,1) G M(W, G) x (Aut(G) x TOP(W)). This is a right translation by a~^ on the first factor of G x W so that r{a){x, w) = (jc •fl~^ It;), and the subgroup of all such right translations is denoted by r(G) C M(W, G). Let 1(G) denote the group of left translations on the first factor so that l(a)(x, w) = (a -x,w). Then elements of €(G) are of the form i(a) = {a-\ti(a),

l) G M(W, G) x (Aut(G) x TOP(W)),

where /x(fl) G Inn(G) is conjugation by a. REMARK 2.5.4. If W is a smooth manifold and we take T O P G ( G xW)n Diff(G x W), then this coincides with the weakly G-equivariant diffeomorphisms of G x W and

DiffG(G X W)=C(W, G) X (Aut(G) x Diff(W)), where C(W,G) is the group of smooth maps of W into G.

2.6. Example: 3-dimensional Seifert manifolds with base the 2-torus We take W = M^. A group Q = 7? acts on W as translations. Let

\^Z->7t

-^ Q-> 1

be a central extension of Z by Q. Then n has a presentation TT = (a, ^, y I [a, ^] = yP. [a, y] = [^, y] = l), where y is a generator of the center Z, and the images of a, fi in Q are generators of Q. Suppose p^O. Using Z c M, one can obtain an effective action of TT on the product R x W as follows: For (z, x,y) eRxW, a(z,x,y) P{z,x,y) y(z,x,y)=

= (z + y,x-\-l,y), = (z,x,y-\-l), (z+

-,x,y\.

Notice that these maps are of the form (z, X, y) ^ (0(z) - X{h(x, y)), h(x, y)),

(2.1)

646

K.B. Lee and F. Raymond

where 0 is an automorphism of M and h is an action of 2 on W, and A is a map H^ ^- R. In our case 0 is always the identity automorphism. Consequently, the group 7t lies in T O P M ( R X W) as

(A, 0, h) e M(W, R) xi (GL(1, R) x TOP(W)). The action of :/r on R x W described above can be explained differently as follows. Consider the Heisenberg group

A^:

I X Z 0 1 y 0 0 1

:

x,y,ze.

which is connected, simply connected and two-step nilpotent. We denote such a matrix by (z,x,y) so that I X z 0 1 y 0 0 1

(z,x,y).

Then the group operation is (z\ x\ y'). (z, X, >;) = a ' + z + x'y, x' + x, / + y), and the center of N is 1-dimensional Z = R, consisting of all matrices with x = y = 0. The quotient W = N/Z is isomorphic to R^ so that 1

N^R^

=

W^l

is an exact sequence of groups. As spaces, this is a smooth fibration which is also a product N = Rx W. Suppose p^Oin the presentation of TT. Let Of = (0,1,0), )g = (0,0,1), andy = (l//7,0,0)GA^. Then these satisfy the relations of the group n so that n sits in A^ as a discrete subgroup, and furthermore, the action of n defined above is nothing but the left multiplication on N by elements of TT. Actually this means that TT \ A^ is a nilmanifold which is a matter that we shall explore more later. The free action of n commutes with the R action on R x R^ and R Pi TT is the 1 0, n 0). A^ The ^u^ R TO)action o^f;^«on^« IP ., i^2 descends to an central Z subgroup generated by y = (-^, = T^ effective 5* = (y) \ R action on TT \ (R x R^). The map TT \ (R l^) ^Z^\R^ is the orbit mapping of a free S^ action. It is the principal 5^-bundle over T^ with Euler class —p.

Seifert manifolds

647

2.7, Injective Seifert constructions In Section 2.5, we described the structure of the group of weakly G-equivariant self homeomorphisms of G x W as T O P G ( G X W ) = M(W, G) X (Aut(G) x TOP(W))

and showed that it was the normalizer of I (G) in TOP(G x W). The action of M( W, G) x (Aut(G) X TOP(W)) on G X W is given by (A,, a, h) • (x, w;) = (Qf(x) • {X{hw)) We observed that i{G) embeds in T O P G ( G la^

(a"^/x(a),

1)GM(W,G)

, hw).

X W)

as

x I n n ( G ) c T O P G ( G x W),

where Inn(G) denotes the inner automorphisms of G. Both Inn(G) and M(W, G) x Inn(G) are normal in M(W, G) x Aut(G) and in T O P G ( G X W), respectively. We may therefore rewrite T O P G ( G X W) as the group extension 1 -^ M(W, G) X Inn(G) -^ T O P G ( G X W) ^ Out(G) x TOP(\y) -^ 1 where Out(G) = Aut(G)/Inn(G). In Section 2.4, we described the condition that must be satisfied if a discrete subgroup n of T O P G ( G X W) is to yield an injective Seifert fibering. Suppose: 1. G is a Lie group. 2. r is isomorphic to a discrete subgroup of G. 3. W is an admissible space (see Section 2.5). 4. |2 is a discrete group acting properly onW Vm p.Q^^ TO?{W). 5. 1 — > F ^ 7 r ^ 2 - > l i s a n extension. DEFINITION.

A homomorphism of TT into

T O P G ( G X W)

so that the diagram of exten-

sions 1

^r

^7T

^Q

^1

\—^M{W,

G) X I n n ( G ) ^ T O P G ( G x W ) - ^ O u t ( G ) x TOP(W)—^1

commutes is called an injective Seifert Construction. The group T O P G ( G X W) is called the universal group for this Seifert Construction. It follows that the group i{G) - 0(n), generated by 1(G) and ^(TT) in T O P G ( G X W), acts properly on G x W. The construction then yields the injective Seifert fibering

an

\ HG) —^ 0(n)

with the typical fiber r\G

\(GxW)^Q\W

and base Q\W.

648

K.B. Lee and F. Raymond

Note that the total space of our Seifert fibering is n \ (G x W) = Q \ ((F \ G) x W). We say it is modelled onG xW. The base space is denoted hy B = Q\W.

2.8. The singular fibers PROPOSITION 2.8.1. Each singular fiber is the quotient of the homogeneous space by a finite group ofajfine diffeomorphisms.

r\G

PROOF. There is an intermediate fibering F \G -> (F \G) x W ^ W so that the following diagram is commutative:

-^ GxW

r\G

—

-^ W

I-

iF\G)xW

-^ w Q\

Q\\

r\G

fibering

^ 7t\iGxW)

^ Q\W

=B

Seifert fibering

The typical fiber of the Seifert fibering

r\G

B

is the homogeneous space F \G. Then, what are the singular fibers? (1) Pick w eW over b e B. (2) Over w, find (F \G) x w in (F \G) x W. (3) Let Quj be the stabilizer of w. Then p~^(b) = Qw\ ((T \G)x w). Another way: (4) Over w, find G x w inG xW. (5) Let Ttuj be the subgroup of n leaving G x w invariant. That is, 1 ^- T ^- TT^ Qu; -> 1 is the pullback of I -> F -^ n ^^ Q ^ I wisi Qw ^-> QThen p-\b) = Quj \ ((F \G) x w) =:nuj\(G x w). The diagram (of the Seifert fibering) restricted to G x u; is -^ G X w

1 F\G

F\G

^ (F\G)xw

{r\G)xw

TtyuMG X W)

-^ w

1^

^ w

Seifert manifolds

649

We need to understand the action of Q^j on (F \ G) x w;, or equivalently, the action of TZU^ onG X w. The typical fibers, which are homogeneous spaces F \G, occur over ^ G 5 for which Qw = ^\ that is, where Q acts freely. For example, if Q acts effectively on W and W is a connected manifold, then the set of typical fibers will be open and dense in E. Each of singular fibers is a quotient of F \ G by a finite group of affine diffeomorphisms of G. This is because when one restricts G x W to G x u;, M(u;, G) xi (Aut(G) x TOP(if)) becomes r(G) xAut(G). Our group F injects into 1{G) and 71^ goes into M(u;, G) x Aut(G) which is the same as r(G) X Aut(G). But r(G) x Aut(G) = i{G) x Aut(G) = Aff(G), because {a,a) = (a, iJi(a)~^) o (1, /x(a) o Of) in M(W, G) x Aut(G). Therefore, we have 1

^ F

^ ^w

I

^ Qw

^ 1

1 — ^ €(G) — ^ Aff(G) — ^ Aut(G) — ^ 1 Thus, IX w acts o n G x u ; c G x W a s a group of affine diffeomorphisms of G. The singular fiber p~\b) = nw\(G x w) is an infra-homogeneous space (see Section 2.9) if the finite group Qw =^w/F acts on F \ G freely, and is an orbifold covered by F \ G otherwise. D 2.9. Infra-homogeneous spaces For a discrete n C Aff(G) for which n acts properly on G and TT fl G = T is of finite index in TT, we obtain an example of an injective Seifert fibering by choosing W = {w}, a point. We get 7t\{Gx{w})^B

= {w}.

This is an injective Seifert fibering with only one fiber, which is singular, and with typical fiber the homogeneous F \(G x {w}). We like to make more precise the geometry carried by the singular fibers of our Seifert fiberings. Endow G with the linear connection defined by the left-invariant vector fields. Since the parallel transport is the effect of the left translations on the tangent vectors of G, and hence clearly independent of paths; the connection is flat. A geodesic through the identity element ^ G G are the 1-parameter subgroups of G and thus defined for any real value of the affine parameter. All geodesies are translates of geodesies through e and thus the connection is complete. One easily checks that the torsion tensor has vanishing covariant derivative. According to [24, Proposition 2.1], Aff(G)=£(G)x Aut(G) is the group of affine diffeomorphisms (= connection-preserving diffeomorphisms) of G, and (a, a) e G x Aut(G) acts on G by (a, a)(x) = a • a(x) for all x GG. For example, if G = R\ Aff(R^) = R'^ X GL(w, R), the ordinary affine group of R^

650

K.B. Lee and F. Raymond

Suppose TT is a discrete subgroup of Aff(G) and acts properly on G. Suppose P CnHG and Q = 7t/r is finite. If n acts freely on G, we call F \ G a homogeneous space and the quotient n \G = Q\(r \G) an infra-homogeneous space. However in general, n will not necessarily act freely and TT \ G is then an orbifold (i.e., a V-manifold). One should notice that the finite regular (possibly branched) covering r\G-^n\G is extremely nice since the action of Q came from Aff(G). The case where G is a simply connected Abelian or nilpotent Lie group is especially important for us. For G = W, 0(n) is a maximal compact subgroup of GL(«, R). A uniform (= cocompact) discrete subgroup TT of R" x 0(n) is called a crystallographic group. By a theorem of Bieberbach, TT Pi R" is isomorphic to IT, and is a lattice (= uniform discrete subgroup) of R". [In general, a lattice F of a Lie group G is a discrete subgroup such that r\G has finite volume. For a solvable Lie group G, this is equivalent to F \ G being compact.] If 7t is torsion-free, we call n a Bieberbach group (= torsion free crystallographic group). Flat manifolds are the orbit spaces TT \ R", where TT is a Bieberbach group. Note that each flat manifold is finitely covered by a flat torus Z'^ \ R". Let G be a connected, simply connected nilpotent Lie group. Choose a maximal compact subgroup C of Aut(G). A uniform discrete subgroup TT of G x C is called an almost crystallographic group. If it is torsion-free, it is called an almost Bieberbach group. An almost Bieberbach group n yields an infra-nilmanifold n \G. Note here again that any infra-nilmanifold is finitely covered by the nilmanifold T \ G, where F = 7T HG. See [39] and [2] for more details. Thus, nilmanifolds are a generalization of tori, and infra-nilmanifolds correspond to flat manifolds. It is also known that a manifold is diffeomorphic to an infra-nilmanifold if and only if it is almost flat. This term is due to Gromov. See [17] for a proof of the above fact.

3. Group cohomology 3.1. Introduction The injective Seifert Construction entails embedding a discrete group 7t into the universal group T O P G ( G X W) in such a way that the following diagram of short exact sequences of groups commutes: 1

>r

^TT I

1 -^

0

M(W, G) X Inn(G) - ^ T O P G ( G XW)-^

^Q

1^^

^1

\(p^p

Out(G) x TOP(W) —^ 1

Therefore, in order to understand when such constructions are possible and to classify them when they do exist, we need to investigate the general procedure for mapping one short exact sequence of groups into another. These results will be formulated in terms of the first and second cohomology of groups with not necessarily Abelian coefficients. We will need the full generality of these results later as we build in more geometry to our Seifert Constructions by "reducing" our universal groups T O P G ( G X W).

Seifert manifolds

651

First, we shall recall some definitions and elementary properties of group extensions. Then we explain the correspondence between congruence classes of extensions of Abelian group A by a group Q in terms of the second cohomology of Q with coefficients in A, if 2(g; A). If A is replaced by a non-Abelian group G, we show that the congruence classes of extensions of G by Q, Opext(G, Q,(p) is in one-one correspondence with H^{Q\ Z{G)), where Z{G) is the center of G. Next, we study the problem of mapping homomorphically one short exact sequence of groups into another. Given one such homomorphism ^o, we show that all other possible homomorphisms are measured by the first cohomology group. Finally, if we change the extension of the initial exact sequence (measured by a second cohomology group) we determine when it too maps into the target exact sequence.

3.2. Group extensions A group extension is a short exact sequence

\^G->

E^

Q-^ 1

of not necessarily Abelian groups. There is a naturally associated homomorphism (p'.Q^^ Out(G), called the abstract kernel of the extension. This comes about as follows. Pick a "section" s.Q-^Eso that the composite Q -^ E ^^ Q is> \ht identity map {s is not necessarily a homomorphism). We pick ^(1) = 1. This defines a map (p'.Q^- Aut(G) (which is a lift of the abstract kernel (p) given by (p{a)=^i{s{a)), where /x is the conjugation map. Even if (jp is not a homomorphism, it induces a homomorphism (p.Q^^ Out(G), which is our abstract kernel. Of course, (p does not depend on the choice of the section s. We say the group E is an extension associated with the abstract kernel (p:Q^Oui(G). Define / : Q x Q ^ G by s(a)^siP)

=

fia,fi)'S(afi).

Then one can easily verify that cp{a) o ^iP) = /z(/(a, P)) o (p{aP),

(3.1)

/(a,l) = l = /(l,^),

(3.2)

/ ( a , P). / ( a ^ , y) = (p(a){f(P, y)) • f(a, py)

(3.3)

for every a,p,y e Q. In fact, (3.1) follows from the definition of the map / above, (3.2) follows from ^(1) = 1, and (3.3) is a result of the associative law of G. Conversely, suppose we have maps (p:Q ^^ Aut(G) (not necessarily a homomorphism) lifting a homomorphism (p:Q ^ Out(G) md f: Q x Q -^ G satisfying (3.1), (3.2)

652

K.B. Lee and F. Raymond

and (3.3). Then there exists an extension \^^G^^E-^Q->\ kernel cp. In fact, E = G x Q as a set, and has group operation

with the given abstract

(a, a). (b, ^) = (a . (p{a)(b) • / ( a , fi), a^). Thus we have PROPOSITION 3.2.1. For a given abstract kernel ^:Q ^^ Out(G), pick a lift (p: Q -^ Aut(G). Then the set of all extensions with abstract kernel (p is in one-one correspondence with the set of all maps f : Q x Q^^ G satisfying (3.1), (3.2) and (3.3).

3.2.2. Let (^: 2 ^ Out(G) be a homomorphism, (p\Q^ Aut(G) be a lift, and f :Q X Q ^^ G a map satisfying (3.1), (3.2) and (3.3). Then the group E = G x Q with group operation DEFINITION

(a, a) . (b, y6) = (a • ^(a)(b) • f(a,

P),ap).

is denoted by G X(^<^) Q. Note that some abstract kernel (p: Q^^ Out(G) may not have any extension at all. When there is a lift (p which is a homomorphism (for example, when G is Abelian), there exists at least one extension (by taking / = 1, the constant map), the semi-direct product G X(i (^) 2 . As a set, this is G x Q, and the group operation is (a,a)'(b,P) DEFINITION

=

{a-(p{a)(b),afi).

3.2.3. Two extensions

1 - > G ^ £ ^ Q ^ 1

and

l ^ G ^ E ' ^ Q ^ l

with the same abstract kernels are congruent if there is an isomorphism 0 : E ^^ E^ which restricts to the identity map on G, and induces the identity on Q. That is, the diagram 1 -- ^ G -- ^ E — ^ Q -—>- 1 '' 1 --^ c

e ' -^ E:' ^

' i2 -—>-

is commutative. DEFINITION 3.2.4. Let <^: 2 ^ Out(G) be a homomorphism, and (p:Q-^ Aut(G) be a lift of (p. Then Opext(2, G, cp) denotes the set of all congruence classes of extensions of the group G by Q with operator cp. Therefore, an element [/] e Opext((2, G, (p) is represented by an extension 1 -^ G ^ • E^ Q^l with E = G X(/,^) Q.

Seifert manifolds

653

3.3. Fullback and pushout 3.3.1. (Pushout) Let T be a subgroup of G. We say (G, F) has UAEP (unique automorphism extension property) if every automorphism of F extends to an automorphism of G uniquely. For example, if G is a connected, simply connected nilpotent Lie group, and F is any lattice, then (G, F) has UAEP. In particular, (R'^, Z'^), where V spans E", has UAEP. Let ^: F ^> G be an injective homomorphism. Suppose (G, s{F)) has UAEP, and 1

1

is exact. Then there is a unique group E fitting the commuting diagram Q

^ G —^ E

Q

-^ 1

pushout by 8 : F ^ G

Such a group E can be constructed as follows. Suppose [/] e Opext(|2, F, (p) is represented by the extension n ^ ^(f,(p) Q' That is, TT = F X 2 as a set, and has group operation (a, a) • (b, P) = {a^ (p(a)(b) • / ( a , P), a^) for a,b e F and a, fi e Q. Recall that (p and / satisfy the equalities (3.1), (3.2) and (3.3) in Section 3.2. Since (G, F) has the UAEP, (p: Q -> Aut(F) can be viewed SiS cp: Q ^^ Aut(F) -^ Aut(G) a n d / : ( 2 x 2 - ^ r a s / : ( 2 x ( 2 - > r - ^ G . With these new interpretations of (p and / , the equalities (3.1), (3.2) and (3.3) still hold. Now we can define E by E = G y<{f\(p) Q- That is, £" = G x 2 as a set, and has group operation (a, ot) . (^, )6) = (a . q>(a){b) • / ( a , for a,b eG and a, p e Q. When e is an obvious inclusion i: F GTT so that 1 — ^ r — ^ T r —^Q—^ n i '' ' TT -^ i1 1 —^ c7 — ^ G 3.3.2. (Pullback) Let 1 ^ G ^ P -^ P ^ 1

P),a^)

G, we denote such an extension E simply by

^

pushout by i'.F —> G

K.B. Lee and F. Raymond

654

be exact, and p : Q ^- P be a homomorphism. Then there is a unique group Q fitting the commuting diagram 1 —^ G —^ Q —^ Q —^ 1

pullback by p:Q^

P

1 —^ G —^ p —^ P —^ 1 Such a group g can be constructed as follows. Suppose [/] G Opext(P, G, (^) is represented by the extension P = G X(j^) P. That is, P = G x P as a set, and has group operation (a, a). {b,P) = {a-^(a)ib)

•

fia,P),ap),

where (p\P -> Aut(G) and f: P x P -^ G. Now, (p: P ^^ Aut(G) gives rise to <^: Q ^• P -^ Aut(G), and / : P X P ^ G gives rise to / : 2 x 0 ^-4^ P x P -^ G. With these new interpretations of ^ and / , the equalities (3.1), (3.2) and (3.3) still hold. Now we can define 2 by, Q = G X(y^^) Q. That is, 2 = G x 2 as a set, and has group operation (a, a) . (b,P) = {a'(p(a)(b) •

f(a,P),aP)

for a, b eG and a, p € Q.

3.4. Extensions with Abelian kernel A and H {Q\ A) Consider the case where G = A is Abelian in the previous section. In other words, we try to classify all group extensions of an Abelian group A by Q with abstract kernel (p:Q ^^ Out(A). Since A is Abelian, Aut(A) = Out(A), and there is no need of having a lift (p. Moreover (3.1) holds automatically. A map f: Q x Q -> A satisfying the two equalities (3.2) and (3.3) is called a factor set (or 2-cocycle) for the abstract kernel (p. The set of all factor sets is denoted by 2 ^ ( 0 ; A) or simply, by Z^(Q; A). Let A,: 2 -> A be any map such that A(l) = 0. Define 8X:Q x Q^^ A by 8X{a, P) = Xia) + cp(a){X(p)) - X{aP).

(3.4)

It turns out that such a 5X is a 2-cocycle and is called a 2-coboundary. The set of all 2-coboundaries is denoted by B^(Q; A). Clearly, J 5 ^ ( 2 ; A) is a subgroup of Z^(g; A). Let / i , / 2 : 2 X (g ^- A be 2-cocycles. We say f\ is cohomologous to /2 if f\ — fi ^ B^{Q\ A); that is, if there is a map X.Q^^ A such that f2 — f\ = 8X. We define the second cohomology group as the quotient group H^(Q;A) =

zl(Q;A)/Bl(Q',A).

Seifert manifolds

655

Since the abstract kernel cp'.Q ^^ Out(A) lifts to a homomorphism (p: Q -^ Out(A) = Aut(A), there is always an extension; namely, the semi-direct product A y\ Q. Then every 2-cocycle f e Z^(Q; A) gives rise to an extension of A by Q, which can be denoted by A ^(f,(p) Q- Furthermore, it is easy to see that two cocycles f\ and /2 yield congruent extensions A X(/j ,^) Q and A y<(f2,(p) Q if and only if f\ and /2 are cohomologous. Consequently, we have Opext(A,e,^)^i/2(e;A) as sets, and they classify the extensions of A by 2 with abstract kernel (/?, up to congruence. It is customary to identify the zero element of H^(Q; A) with the extension class of the semi-direct product A xi Q. Then addition of the 2-cocycles in Z^(Q, A) induces the group operations in H^(Q; A).

3.5. Extensions with non-Abelian kernel G and H^{Q\ Z{G)) For any homomorphism (p: Q ^^ Out(G), take any lift (p: Q ^^ Aut(G). This, in turn, induces a unique homomorphism (p: Q ^^ Aut(Z(G)) by restriction (regardless which lift of (p is chosen, and even if cp was not a homomorphism). We shall define an action of H^(Q; Z{G)) on the set Opext(G, Q; cp) which will turn out to be simply transitive. Let g G Z ? ( 2 ; Z ( G ) ) ; that is, g:Q x 2 -> Z(G) satisfying the equalities (3.2) and (3.3) of Section 3.2. For any [/o] e Opext(G, Q; cp), define

Then, clearly, g • /o satisfies the equalities (3.1), (3.2) and (3.3). Moreover, if two extensions G X(yQ^) Q and G X(j,^) Q are congruent, so are G X(^.yQ ,^) Q and G X(^.y^, ,^) Q. Therefore (g, /o) i-> g • /o is an action of Z^(Q; Z(G)) on Opext(G, Q; cp). Suppose [g ' f] = [/] for some [/] e Opext(G, Q; cp). Then one easily sees that g e B^(Q; Z{G)). This gives us a simple action of H^{Q\ Z{G)) on Opext(G, Q\ cp). Now we show that this action is transitive. Suppose / , / ' e Opext(G, Q; cp). Let

for all a, P e Q. Then g(a, P) lies in Z(G), and it is easy to check

g:QxQ^Z(G)

656

K.B. Lee and F. Raymond

satisfies the equalities (3.2) and (3.3). Thus, g

G Z^~{Q\ Z(G)).

Consequently, we have

shown the action of H^Q; Z{G)) on Opext(G, Q\ cp) is simply transitive. Thus, Opext(G,e,^)^//?(G;^(G)) as sets, and they classify the extensions of G by 2 with abstract kernel (p, up to congruence. In general, however, there may not exist any extension of G by g with the given abstract kernel. Moreover, even when there exists one, there is no a priori special element, like the semi-direct product G xi 2 , because the semi-direct product can be formed when and only when the abstract kernel lifts to an homomorphism into Aut(G).

3.6. H\Q;

C) with a non-Abelian C

The first cohomology is useful in classifying homomorphisms from one group to another. Let C be a group (not necessarily Abelian) and (p.Q^- Aut(C) be a homomorphism. We shall define the first cohomology set i/^ ( 2 ; G). A map r]\Q^^ C is called a \-cocycle if it satisfies

for all Of, ^ G Q. The set of all 1-cocycles is denoted by Zl,(2; G). Two 1-cocycles T], ri^: Q ^^ C arc cohomologous if there exists c eC such that r]\a) = c • r](a) • (p(a)(^c~ ) for all a G 2- The set of cohomologous classes of ZJ,(2; G) is denoted by When G is Abelian, //^ ( 2 ; G) is the ordinary first cohomology group.

HhQ\C).

3.7. Mapping one group extension into another DEFINITION

3.7.1. A diagram of short exact sequences of groups

1 —^ G

^ E ^

^ Q —^ 1 e

1 — ^ G' — ^ E' — ^ Q' — ^ 1 is called half-commutative if / is an inclusion and (A) the right-hand side square is commutative, and (B) for every X G £", i{x -a - x~^) = \l/(x) • i(a) • Vr(jc)~^ for all A G G .

(3.5)

Seifert manifolds

657

Condition (B) is weaker than the left-hand side square being commutative (i.e., \lr\Q =i).\i simply means that / o /x(x) = /x^(Vr(x)) oi\G -^ G' for all X G £". It is not required that '^\Q = i. Let [/] € Opext((2, G, (p) represent the congruence class of the extension E. That is, E = G X (j^) Q. The homomorphism ^l/'.E^^E' gives rise to a map

f :e^Aut(GO by (p\a) = /x'(Vr(l,Qf)), where ii' is conjugation in E'^ Since (p{a) = /X(1,Q;), the two maps cp: Q ^^ Aut(G) and (f^: Q -^ Aut(GO are related by the commuting diagram

G' — ^ G' This shows that (p'{a) maps /(G) onto itself, and hence C = Cc'ii^G)), the centralizer of /(G) in G', onto itself also. THEOREM 3.7.2. Let \l/: E -^ E' be a homomorphism which makes the diagram (3.5) half commutative. Let E = G y<(f,^) Q, cind define (p'{ot) = /x^(i/^(l, a)) for a e Q {ii' is conjugation in E'). Then, (1) There exists a homomorphism 0 '.E -> E' making the diagram

—>^G -—^E>i ^ G '

--^

- ^ Q - -^ 1

'0 E' -

(3.6)

' '^ y -- ^ 1 ^ k >

iT

commutative if and only if there exists a map X: Q -^ C = i{f(a, ^ ) , 1) . Xia^r^

= X(ay^ • (p\a){X(pr^)

CG'(/(G))

satisfying

• iA(/(a, P), l)

(3.7)

for alia, P e Q. (2) Suppose that there exists a homomorphism OQ: E -^ E' as above. Then p.' o ^o ii^duces a homomorphism (p" '.Q^- Aut(C). The set of all 0 's fitting the diagram, up to conjugation by elements ofG' {in fact, ofC), is in one-one correspondence with HK(Q; C). (3) Suppose that there exists a homomorphism Oo'.E-^E'as above, and that i(Z(G)) C Z{G'). Let [E"^ e Opext((2, G, (p) be an extension. Then there exists a homomorphism E" -^ E' completing the diagram if and only if [£"] - [E] e Ker{/,: H^{Q- Z{G)) -^ H^{Q;

Z(G'))].

658 PROOF.

K.B. Lee and F. Raymond

(1) The extension £" is G x 2 with group law (a, a) • (Z7, ^) = {a . ^{a){b) • / ( a , )6), ap).

Then ( l , a ) ( a , l ) ( l , a ) - ^ = (^(a)(a),l). Suppose there is a homomorphism 0:E^^E'

making both squares commutative. Then

= /((l,a)(a,l)(l,cy)-i) = i/r(l,a)./(«,l)-V^(l,Qf)-^ Since conjugations of / (a, 1) by ^(1, a) and i/r(1, of) are the same, their difference must he in the centrahzer of / (G) in G^ In other words, there exists a map X:Q^C

= CGii{G))

so that 0(\,a) = X(a)~^ • V^(l, of). Thus, in general, O(a,oi) =

0(a,l)'0(l,a)

= i(a,l)-X(a)-^

-i^(l,a).

Now ^((l,a)(l,^))

=0{f(a^P),aP)) = i{f(a, P), 1) . X(aP)-^ . ifihafi).

(3.8)

On the other hand, 0(ha).

0(hP)

= (A(a)-i . xlfiha)) . (M)^)-^ • ilf(hP)) = A(«)-'.{^(l,a).A(^)-^iA(l,cy)-i}.{iA(l,«)-^(l,)S)} = M a ) - ' . f (a)(A(;6)-').iA(/(c^,)^),c^)^) = X(a)-' . f (a)(A(^)-'). ilf{f(a, ^ ) , l) • ^ ( 1 , a^S).

(3.9)

From the equalities (3.8) and (3.9), we get i{fia, ^ ) , 1) . Hafi)-' = Xia)-^ • ^\a){X(P)-^) • I A ( / ( « , P), l). Conversely, suppose there exists X: Q ^^ C satisfying this equality. One simply defines 0:E-^ E'by e(a,a) = i(a, I) • Xia)'^ -1^(1,0?).

Seifert manifolds

659

We claim that ^ is a desired homomorphism. This is shown by a series of calculations: 6>(a,a).6>(Z?,)6) = {i{a, 1) • X{a)~^ . iA(l, a)} • [i{b, 1) • A.(^)"^ • 1^(1,^^)} = i(a, 1). X{a)-^ . [f{\,ot).

(definition of 6>)

/(Z7,1) • iA(l, a)-^}

. {Vr(l,a). A(^)-* . iA(l,a)-^} • {iA(l,a). iA(l,^)} = /(a, 1) • A(a)-l . ^\a){i{b,

D) . (^'(a)(A()g)-^) • V^(/(a, ^ ) , a^S)

(since V^ is a homomorphism and (1, Qf)(l, ^) = (/(a, P), afi)) = i(a, 1) • (p\a){i(b, D) . A(a)-' • (p\a){X{P)-^) • ^^(/(a, ^ ) , l) • xlrihafi) (M«)-'GC)

= /(a,l)-/(^(a)(Z7,l)).{A(a)-^^\a)(M^)-')-V^(/(a,)6),l)}.iA(l,a^) [(p\a) oi = i o^(a)) = i(a, l)'i{cp(a)(b, D ) . {/(/(a,^), l) .Ma^)-^} • iA(l,«i^) (equality (3.7)) = /(a . (^(a)(/7). / ( a , ^)) • A(a)0)-^ • = 0{a- ^(a)(b). /(Of, p), afi) = 0[(a,a)' (b, P))

^ihafi) (definition of ^) (group operation in E).

(2) For X e E,c eC = Cc'iiiG)) and a G G, it is easy to see that /ji\Ooix))(c) and i(a) commute with each other. This shows that the homomorphism E —> E' —> Aut(GO, where JJL' is conjugation in E', leaves C invariant. Therefore we have a homomorphism E ^ E ' ^

Aut(C).

Furthermore, for a G G,

{^l'oe^){a) = ^l'{i{a)) is trivial on C. Thus E -^ Aut(C) factors through Q. We have obtained a homomorphism (^^^ = /x^o6>o:!2^ Aut(C). (Note that (p" is actually a homomorphism, even though (p'.Q^^ Aut(GO is not.) Let ^ : £• ^ £•' be a homomorphism fitting the diagram. Then 0 must be of the form

for some map r;: £" ^- G^ It is easy to verify that r) satisfies r^iaP) = 11(a) • OoiaM^Woia)-^

= r](ot) • (p'(a){r](P)).

660

K.B. Lee and F. Raymond

Since 0\Q = i = ^QIG, we have ri{a) = 1 for all a e G. For any a e G and a e E, T](aa) = r](a) from the above equality. Moreover, rj(a) = rj(a • (a~^aa)) = r](aa) = ri(a) • Oo(a)rj(a)Ooia)~^ = 1 • i(a)r](a)i(a)~^ = i(a)r](a)i(a)~K This shows r] has values in the centralizer C = Cc'iiiG)). Thus r]:Q^C

=

CGiiiG)),

and hence rj e ZK(Q; C). See Section 3.6. Conversely, a map rj: Q ^^ C satisfying the cocycle condition gives rise to a homomorphism T] -OQ'.E ^^ E'. Consequently, T] <

> T] • ^ 0

is a one-one correspondence between Z^,((2; C) and the set of all homomorphisms E -^ E' inducing / and ^ on G and (Q, respectively. Let rj,r]' e ZK(Q; C), and suppose there exists c eC such that (ri'-Oo){a)=c^(r]^Oo)(a)-c-^

for alia £ E. This is equal to r]\a) = c • r]ia) • (p'\a){c~^); i.e., W] =

[r]]eHl.(Q;C).

Consequently, HK(Q; C) classifies all homomorphisms E ^^ E' inducing / and ^ on G and 2 , respectively, up to conjugation by elements of C. (3) Let

\^G'^£^Q^\ be the pullback (see Section 3.3.2) of the exact sequence 1 ^- G' ^- f"^ ^ 2 ' ^ 1 via 0 \Q ^^ G^ For any \ ^^ G ^^ E" ^^ Q ^^ 1, there exists a homomorphism E^^ -^ E' making the diagram 1

^ G /

^ E" y

^ Q

^ 1

\e

Y

1 — ^ G' — ^ E' — ^ Q' — ^ 1

Seifert manifolds

661

commutative if and only if there exists a homomorphism E" -^ £ making the diagram 1 -—^ G ' 1 --^

^ E " — ^ Q --^

i

\

1

\^

c;' — ^ e — ^ Q' -- ^ 1

commutative. Suppose OQ'.E ^^ E' exists as in the diagram (3.6). This yields E -^ E. We identify Opext(G,G,<^) ^ Opext(G,e,<^') ^

H\Q',Z(G)),

H^{Q\Z(G')).

Since i(Z(G)) C Z(G'), i: Z(G) -^ Z{G') induces a homomorphism /* : H^{Q- Z{G)) -^

H^Q;

Z(G')),

Then, clearly, [E^'] e Opext(G, Q, cp) can map into [E^] (or, equivalently into £) if and only if [£''] - [E] e H^(Q\ Z{G)) maps to 0 G H^{Q\ Z{G')). D Unless there is one definite extension E and a homomorphism ^o. one cannot conclude that there is a homomorphism E'' -^ E' using this cohomology homomorphism. On the other hand, suppose that there is an extension E and a homomorphism E -^ E\ and that H^(Q; Z{G')) = 0. Then every extension is in the kernel of H^(Q; Z(G)) -^ H^(Q; Z{G')), and hence every extension with the abstract kernel (p can be mapped into E'. 4. Seifert fiber space construction with T O P G ( G

X W)

4.1. Introduction Recall, from Lemma 2.5.3, that the group of all weakly G-equivariant self-homeomorphisms of G X W is T O P G ( G X W) = M(W, G) XI (Aut(G) x TOP(W)), the universal group for a Seifert Construction. Let r C G be a discrete subgroup, p'.Q^^ TOP(W) a properly discontinuous action, and l - > r ^ - 7 r ^ - | 2 - > l a group extension (with abstract kernel (p:Q-^ Out(G)). Our goal is to find a homomorphism 0:7t ^^ T O P G ( G X W) which makes the diagram 1

^r

^n

^Q

^1

(pxp U 1 — ^ M(W, G) X Inn(G) — ^ T O P G ( G X W) — ^ Out(G) x TOF(W) — ^ 1

commutative.

662

K.B. Lee and F. Raymond

The construction of a homomorphism 0 m ^^ T O P G ( G X W) will be achieved in two steps. First, we map n into a group Gn (see Section 3.3.1), and second Gn into T O P G ( G X W). For the first step, we require, in addition, that T be a lattice in a simply connected completely solvable Lie group or a semi-simple Lie group for which Mostow's rigidity theorem holds. Then we stack the two steps together to get our desired homomorphisms. A complete proof for the Abelian case is given. The other cases rely heavily on the Abelian case, and we sketch and reference the proofs for them. As it turns out, the proofs of the existence also leads to uniqueness and rigidity theorems for these homomorphisms. The meaning of existence, uniqueness and rigidity are then explored, for it is these properties on which many of the applications of the Seifert Construction are based. Finally, we observe that the Main Theorem 4.3.2 of this section remains valid in the smooth category. We treat the second step first.

4.2. Mapping an extension ofG into TOFG(G

X W)

Interpreting the algebraic criterion of Theorem 3.7.2 with E = Gn and E^ = T O P G ( G W), we get

X

4.2.1. Let p'.Q^^ TOP( W) be a properly discontinuous action, and 1 ^• G^E^^Q^^lan extension with abstract kernel (p. Suppose [/] G Opext((2, G, (p) represents the extension E. That is, E = G y^{f,(p) Q (see Section 3.2.2). (Then cp: Q ^^ Aut(G) and f \Qy^ Q^- G satisfy the equalities (3.1), (3.2) and (3.3) of Section 3.2.) (1) There exists a homomorphism 0 : E -^ T O P G ( G X W) making PROPOSITION

1

^G

^E

^Q

1^^

^1

oxp

1 ^ M ( W , G ) xInn(G)^TOPG(Gx W)^Out(G) x T O P ( \ y ) ^ l commutative if and only if there exists a map X: Q ^M(W,G) fia, P) = {^(a) o X(P) o p(ar^}

satisfying

• X(a) • X(aP)-^ e r(G) C M{W, G)

(4.1)

for all a, p e Q. [Compare this with the equality (3.4) defining 8X.] (2) Suppose there exists a homomorphism OQ: E -^ T O P G ( G X W) as above. Then li' o OQ {pi' is conjugation in T O P G ( G X W)) induces a homomorphism (p" :Q -^ Aut(M(W, G)). The set of all 0 's fitting the diagram, up to conjugation by elements 6>/M( W, G), is in one-one correspondence with H\,{Q\ M ( W , G ) ) . (3) Suppose that there exists a homomorphism OQ'.E ^^ T O P G ( G X W) as above. Then OoiZiG)) C M(W, Z(G)). Let [E''] e Opext((2, G, cp) be an extension. Then there exist a homomorphism E" -> T O P G (G x W) completing the diagram if and only if [E"^ - [E] lies in the kernel ofU ' H^{Q\ Z{G)) -^ H^{Q\ M(W, Z{G))).

Seifert manifolds PROOF.

663

(1) We apply Theorem 3.7.2(1) to the diagram 1

^G

^E

^Q

i

^1 <^XP

1 ^ M ( W , G) X Inn(G) — ^ T O P G ( G X W) —^Out(G) x TOP(W) - ^ 1 Let V^: £ ^ T O P G ( G X W) be given by if {a) = (1, (^(a), p{a)) e M(W, G) x (Aut(G) x TOP(iy)), where (p{a) e Aut(G) is conjugation in £, and p:E ^^ Q ^^ TOP(W) (abuse of notation) is the given properly discontinuous action of 2 on W. In particular, fil, a) = {I, (p(a), pia)). Conditions (A) in Section 3.7 is easily checked. For (B), Let a eG and a e E. Then l{a'a-a~^)

= =

l{^(a)(a)) {(p(a)(ar\fi{(p(a)(a)),l).

On the other hand, f(a)'lia)'ir(a)-^

= {h(p(a), p(a)) ^ {a-\ ^i(a),l) - {h^(ar\ =

p(a)-^)

{cp(a)(a)-\fji{^ia)(a)),l).

Thus,

as we wanted. Now we examine the equality (3.7).

= {Hapy'

. f(a, pr\

/x(/(c., ^)), 1),

while

= {X{a)-\ 1, 1) . {f{\,a)^k{^r^ = {X(a)-\

. iA(l,c.)"^} • (l,/x(/(a,ye)), 1)

1, 1) . {(1, (p{a), p{a)) • {m)-\

1, l) • (l, q>{a), p(a))~^}

.(l,/x(/(a,^)),l) = ( M a ) - ^ 1, 1) . {(p{a)ok{P)-'op{a)-\ = {X{ar'

1, 1) . ( l , / x ( / ( a , ^ ) ) , 1)

. {^(a) o A(^)-i o p ( a ) - i } , /x(/(a, ^)), l).

664

K.B. Lee and F. Raymond

Therefore, £ ( / ( a , P)) . Xiafi)-^ = Xia)-^ • ^(a){X(pr')

• xlr{f(a, p))

if and only if X(apr^

• / ( a , P)-^ = X(ay^ • {^(a) o X(P)-^ o p ( a ) - i }

or equivalently, fia, P) = {^(a) o XiP) o p ( a ) - i } • X{a) • X M ) - » as elements of G. This is the identity (4.1). (2) Suppose there exists a homomorphism OQ-.E -^ T O P G ( G X W) as above. Since i(G) is normal in T O P G ( G X W ) , and the centraUzer of i(G) in M(W, G) >^ Inn(G) is M(W, G), M(W, G) is normal in T O P G ( G X W). Consequently, conjugation by elements of TOPG ( G X W) leaves M( W, G) invariant, and JJL' O ^Q induces a homomorphism (p'':Q-^ Aut(M(W, G)). The rest is clear. (3) Observe that the center of M(W, G) x Inn(G) is M(W, Z{G)) so that £(Z(G)) C M(W, Z(G)). Now apply Theorem 3.7.2(3). D 4.2.2. When the condition (4.1) is satisfied, vanishing of H\Q; M ( W , G ) ) and H^{Q\ M(W, Z(G))) (e.g., if Z{G) is contractible) yields strong consequences by the statements (2) and (3). Namely, (2) The homomorphism 0 fitting the diagram is unique up to conjugation by elements ofM(W,G). (3) Every extension with the abstract kernel ip can be mapped into T O P G ( G X W). REMARK

The vanishing of H^iQ', M(W, Z(G))) only says that there is at most one extension of M(W, G) X Inn(G) by Q for a given abstract kernel. This alone is not necessarily enough to guarantee that the condition (4.1) is automatically satisfied because the coefficient groups G and M(W, G) x Inn(G) have distinct "operator groups". On the other hand, we have the following important observation. 4.2.3. If^c ' Q - ^ Aut(M(W, G) x Inn(G)) -> Aut(M(W, G)) is a homomorphism, and H^iQ', M{W, Z(G))) = 0, then 0 exists as in Proposition 4.2.1, and consequently condition (4.1) is satisfied. THEOREM

PROOF.

Let

1 —^ M(W, G) X Inn(G)

—^£—^Q—^l

be the pullback of 1 —^ M(W, G) x Inn(G) -^ Out(G) X TOP(W) —> 1. Then £: = (M(W,G)xInn(G))x(/,,^)e

M(W, G) x (Aut(G) x TOP(W)) —>

Seifert manifolds for some maps ^:Q^ satisfying

665

Aut(M(W, G) x Inn(G)) and /z: 2 x g ^ M(W, G) x Inn(G)

(p{a) o ^(p) = /x(/z(a, P)) o (piotP), h(a, P) • h(ap, y) = (p(a){h(P, y)) • h(a, Py) for every a, p,y G Q. Note that the pair (h, (p) is not unique. (In fact, each (p(a) can change by conjugation by an element of M(W, G) xi Inn(G), and h changes accordingly.) The map (jp induces two maps (pc'-Q-^ Aut(G) and cpc'-Q ^^ Aut(M(W, G)). The second map in the definition of (pc is obtained as follows. Since M(W, G) xi Inn(G) = ^(^) XZ(G) M(W, G), and i(G) is normal in T O P G ( G X W), an automorphism of M(W, G) X Inn(G) leaving i(G) invariant induces an automorphism of M{W, G). Also ^G induces a homomorphism cp: Q-^ Out(G). Now let l ^ G ^

E^Q^1

be any extension with abstract kernel (p:Q ^^ Out(G). Then there exists a map f: Q x Q^^ G such that E = G X{f,^Q) Q satisfies i>G(oi) o (pciP) = l^{f(oi, P)) o (PG{OIP).

/ ( a , l ) = l = /(l,)s), /(a, P). f{ap, y) = ^G(a){f(p, y)) • /(a, Py) for every a, P,y e Q. Let us examine the first equality. This equality holds in the group Aut(G). Now f{a,P) e 1(G) and M(W,G) = C centralizes i(G). Hence /x(/(Qf,^)) on C is trivial. If we replace <^G in the first equation by (pc, then the equation is satisfied on M(W, G) in M(W, G) x Inn(G) because /x(/(Qf, P)) is trivial and cpc is a homomorphism. Since i(G) and M(W, G) generate M(W, G) x Inn(G), the first equality still holds in M(W, G) X Inn(G) if we replace ^G with (p. Because (p\i(G)M(-^(g)) = ^( 1(G) C M(W, G) x Inn(G). Consequently, £" = G X(/^^) 2 is sitting in r = (M(W, G) X Inn(G)) X(^,^) Q. Now, if H^(Q; M(W, Z(G))) = 0, the extensions £ and f' are congruent, and we have a desired homomorphism E ^^ £. This theorem will be used to prove Theorem 4.3.2. D

4.3. Construction with T O P G (G

X W)

4.3.1. We shall say a discrete group F is special if F is isomorphic to a lattice in any one of the following Lie groups G: (51) R^ for some )^ > 0, (52) a simply connected nilpotent Lie group.

666

K.B. Lee and F. Raymond

(53) a simply connected completely solvable Lie group; that is, for each X eQ, the Lie algebra of G, ad(X) '.Q ^^ Q has only real eigenvalues, (54) a semi-simple centerless Lie group without any normal compact factors and if G contains any 3-dimensional factors (i.e., PSL(2, R)), then the projection of the lattice to each of these factors is dense. We shall also call the Lie group G special. Such groups possess ULIEP (Unique Lattice Isomorphism Extension Property). That is, any isomorphism between such lattices extends uniquely to an isomorphism of G. Even more generally, if Pi is a lattice in a special G/ (/ = 1, 2), then any isomorphism cp'.Fi -> r2 extends uniquely to an isomorphism ^:Gi-^ G2. Notice that: type (SI) =^ type (S2) =^ type (S3). The following is the main construction. Its proof is deferred until Section 4.5. 4.3.2 ([12,26,31,54]). Let F cG bea special lattice. Let p.Q-> TOP(W) be a properly discontinuous action. Then for any extension I^F^^TT^Q-^I (with abstract kernel cp: Q-> Out(G)) the following are true: (1) Existence: There exists a proper action O'.n -^ T O P G ( G X W) making the diagram THEOREM

1

^F

^TT

I

^Q

^1 (p^p

1 —^M(W, G) X Inn(G) — ^ T O P G ( G X W) —^Out(G) x TOP(W) — ^ 1 (4.2) commutative. For completely solvable G we have in addition: (2) Uniqueness: Congruent extensions are conjugate in M(\y, G) C T O P G ( G X W). More precisely, suppose 0\, 62:71 -> T O P G ( G X W) are two homomorphisms which fit in diagram (4.2) with fixed I and cp y^ p, then there exists A G M(W, G) such that 02 = fi(X)o0i. (3) Rigidity: Suppose 0\,02:IT -^ T O P G ( G X W) are two homomorphisms which fit in the diagram (4.2) (possibly with distinct i and p). Then there exists {k,a,h) e T O P G ( G X W) such that 62 = pQ^.a.h) oO\, provided that p2 = p(h) o p\. Uniqueness and rigidity for the semi-simple case (S4) will be discussed in Sections 4.5.6 and 4.5.7

4.4. The meaning of existence, uniqueness and rigidity 4.4.1. (Existence) Suppose there exists 0:n -> TOPG ( G X W) making the diagram (4.2) commutative. Then 0(7t) acts properly onG xW. This yields the injective Seifert fibering 1(F) \ KG) —^ 0(7t)

\(GxW)^Q\W

with the typical fiber F \G and base Q\W. We have explained the properties of such a fibering, and its singular fibers in Sections 2.7, 2.8 and 2.9.

Seifert manifolds

667

4.4.2. (Uniqueness) Let ^o, ^i be homomorphisms of n into T O P G ( G 1

^r

^7t Y

such that

^Q

Y

1 ^ M ( W , G) X Inn(G) ^

X W)

^1

Y

T O P G ( G X

1^) ^ O u t ( G ) x TOP(W) — ^ 1

is commutative. If there exists A e M( W, G) such that 0{ = /x(A,) o ^o, the map X:W ^^ G induces a homeomorphism of G x W by A(x, u;) = (x • (A(M;))

, w),

see Section 2.5. This, in turn, yields a homeomorphism [A,]: j^r^ -^ f^FT- ^^ ^^^ commuting diagram shows

Y

Y

P(Q)\W

^^p(Q)\W

the map [A] induces the identity map on the base space. In fact, X is G-equivariant (that is, it commutes with the left principal G-action). Such spaces MQ and Mi are said to be strictly equivalent. When G is special of type (SI), (S2) or (S3); or W is contractible, of type (S4) (see Section 4.3.1), A, is homotopic to the constant map ^: W ^- G (^ is the identity element of G). Then the path {A^: 0 < ^ ^ 1} gives rise to a continuous family of homomorphisms (9r:7r^TOPG(Gx W)by

for a G TT, and consequently a continuous family of homeomorphisms [Of]:

GxW

GxW

00(7t)

0t(7t)

Thus jjp^ can be deformed to j4^ by moving just along the fibers. In fact, the family A / i G x W - ^ G x W i s G-equivariant. If ^o(7r) commutes with 1(G), then the deformation [Ot] is G-equivariant. 4.4.3. (Rigidity) Let ^o, 0\ be homomorphisms of TT into T O P G ( G 1

^r

^7T

such that

^Q

^1 bixpi

^i

1 —^M(\y, G) X Inn(G) - ^ T O P G ( G

X W)

X

W)^OuiiG)

x TOP(W) —^ 1

668

K.B. Lee and F. Raymond

is commutative. If there exists (X,a,h) e TOPG(G^ X W) such that 0\ = /jL(X,a,h) the map (k,a,h):G x W ^^ G x W induces a homeomorphism [X,a,h] A4 — G><W l^^aM GxW _

Po{Q)\W

^ ^

OOQ,

,^,

Px{Q)\W

which preserves the fibers. More precisely, [A, a, /i] is defined by [A, a, /i]([jc, w]) = [(A, a, h)(x, w)]. Since (X,a,h)oOo(a) = 0\(a)o(X,a,h) for all a e TT, the map [X,a,h] is well-defined. Further, (X,a,h)(x,w) = (a(x) • Xh(w), h{w)) shows that [k,a,h] is a map from Mo to M\. That is, [A,, a, /z] is the descent of the weakly equivariant fiber preserving map (k,a,h) sending G fibers to G fibers. Such spaces MQ and M\ are said to be equivalent. The conjugation of ^o by (A, a, h) is called a Seifert automorphism of G x W. The induced homeomorphism [A, a, /z]: MQ -^ M\ is called a Seifert equivalence or Seifert automorphism. If /^ is characteristic in n, any automorphism of n induces an automorphism of F and Q. Consequently, if MQ and M\ are Seifert fiber spaces modelled on G x W which have the same fundamental group (or "orbifold fundamental group") they are equivalent as Seifert fiber spaces, provided that the base spaces are rigidly related (i.e., there exists h e TOF(W) for which pi = IJLQI) O po). Similarly, if r :7r ^^ TT Ms an isomorphism carrying F isomorphically onto F\ and inducing an isomorphism f: Q ^^ Q', then the analogous rigidity statement for M = 0(n) \(G X W) and M' = O^n') \ (G x W) holds, cf. [12, 8.5] and [26, 2.4]. More precisely, if there exists h € TOF(W) such that /x(/z) o p = p^ o f, then there exists h = {k,a,h)eTOPG(G x W) such that 6>'o r =/x(^) o6>. EXAMPLE

4.4.4 (When W is a point). Suppose W = {/?}, a point. Then

T O P G ( G XW)

= M(p, G) X (Aut(G) x TOF(p)) = r(G) X Aut(G) = i(G) X Aut(G).

Also, note that M(p, G) x Inn(G) = r(G) x Inn(G) = 1(G) x Inn(G). Since Q acts on W properly, Q must be a finite group. We assume that F C G is a special lattice, and apply Theorem 4.3.2. Then diagram (4.2) becomes 1

^ r

^ TT 0

^ Q

^ 1

(p

1(G) X Inn(G) — ^ i(G) x Aut(G) — ^ Out(G) — ^ 1

Seifert manifolds

669

Furthermore, since t{r) C i{G), the above diagram induces 1

^ r

^ n

^ Q

I

^ 1

H>

1 — ^ €(G) — ^ ^(G) X Aut(G) — ^ Aut(G) — ^ 1 where (^ is a homomorphism Ufting ip. [In particular, this shows that, for special G, the abstract kernel (p\Q-^ Out(G) (with Q finite) lifts to a homomorphism (p'.Q-> Aut(G).] Consequently, 0(7t) C i(G) xi C for a finite subgroup C C Aut(G). This means that there exists a Riemannian metric on G for which 0{7i)
4.5. Proofs of Theorem 4.3.2 There are two steps. First we map n into an extension £" of G by Q, and second we map E intoTOPc(Gx W). 4.5.1. Every special Lie group G has the ULIEP. For every G with ULIEP, let 1 ^ F -^ TT -^ 2 ^- 1 be an extension which represents / G Opext(F, Q, (p) satisfying (3.1), (3.2) and (3.3) of Section 3.2. One can form a pushout by the inclusion i: F ^^ G to obtain E = Gn. See Section 3.3.1. Then / e Opext(G, Q, (^) and the diagram 1 —^ r

^ n

Y

Y

^ Q —^ 1 Y

1 — ^ G — ^ E = Gii — ^ Q — ^ 1 is commutative.

670

K.B. Lee and F. Raymond

4.5.2. Now in order to map E into T O P G ( G X W ) , we apply Proposition 4.2.1 for the cases (SI), (S2), and (S3). These cases depend upon the Abelian case whose proof we give in complete detail. When G = R^ is AbeUan, the existence of A in the equality (4.1) / ( a , fi) = {^(a) o X(P) o pia)-^) . k(a) • Xia^y^ means that [/] will have to be 0 in H^iQ; M(W, R^)). The following lemma takes care of this problem: LEMMA 4.5.3 ([12, Lemma 8.4] and [13, Theorem 7.1]). Let p:Q^W be a properly discontinuous action with Q\W paracompact. For any homomorphism (p: Q -^ GL(/:, R), and an action of Q on M{W, R^)bya'k = cp {a) oXo p{pi)-\ W{Q\ M(W, R^)) = Ofor all i > 0. PROOF. We shall prove the lemma, as was done in [12], for g \ W compact. For the general case we refer the reader to [13]. Let Ux be a neighborhood of x eW with the property that QxUx = Ux and such that ifUxf^otUxi^ 0, then a e Qx-l^^iVx — QUxTake M{Vx)
}lomiQ^{ZQ;M(Ux))^M{Vx) because ZQ is a free IJQX module with a basis given by choosing coset representatives. Using Shapiro's lemma, it follows that H''{Qx\M{Ux))^H''{Q\M{Vx)). (See, e.g., [11, Chapter X, Proposition 7.4] or [9, Chapter III 5.8 and 6.2].) Now W {Qx\ M{Ux)) = 0 for all / ^ 1, because Qx is a finite group and M{Ux) is an R-vector space. \f y ^ Q{x), then neighborhoods Ux and U' can be chosen so that V^ n f/^ = 0 because Q acts properly discontinuously on W. This insures that Q \W is Hausdorff (and completely regular). Because 2 \ W is compact, there exists a finite set of points jci, JC2,..., x„ with neighborhoods (/^,, Ux2^ • •, Ux^ as above whose images, y^* , V^*,..., y^* in Q\W cover Q\W. Take a partition of unity subordinate to this open covering. By composing each partition function with the quotient map W —> Q\^. we get a partition of unity 8\,e2, ...,Sn on W subordinate to Vi, V2, • • •, K , where Vi = p'^iVx) = QxiiUxi). Each partition function satisfies 8j{aw) = ej(w), for all a e Q, w e W. Now multiplication by 8j is a Q-module homomorphism of M(W, R^)

Seifert manifolds

671

into M(Vj). Consequently, there is induced homomorphisms Sj^ and the commutative diagram if*(e;M(W,R^)) - ^

/ / * ( e ; M(Vj)) —^^ with £1* + ^2* H

H*(Q;M(W,]

H\Q',

M{Vj))

h 6:«* = identity. This yields the lemma.

D

4.5.4. For a nilpotent Lie group G, one shows that the equation / ( a , P) = {(p(a) o X(P) o pia)-^) . X(a) • A(a^)-i has a solution for A. One reduces this problem to iterated applications of H^(Q; M( W, M^)) = 0 to guarantee the existence while H^{Q\ M(W, R^)) = 0 for the uniqueness. The readers are referred to [26] for the details in the nilpotent case. 4.5.5. Theorem 4.2.3 is used to prove the existence in the completely solvable and type (S4) cases as well as providing a different procedure to that used in [26] for the nilpotent case. Details can be found in [41, Theorem 3]. The uniqueness for the completely solvable case is not given in [41] but it can be obtained inductively from the nilpotent case. Rigidity in the abehan case was first proved by a cohomological argument, [12, §8.5]. In the nilpotent case, rigidity follows from uniqueness and ULIEP, [26, §2.4]. In fact, the same argument works for completely solvable G. 4.5.6. Suppose Aut(G) splits as Inn(G) xi Out(G), and Z{G) = 0 (for example, G of type (S4)). If we replace M(W, G) by r{G) and T O P G ( G X W) by r{G) x (Aut(G) x TOP(W)) C T O P G ( G X W), then a homomorphism Q-.E-^ r(G) xi (Aut(G) x TOP(W)) exists (with ^IG = ^ on G ) by Theorem 4.2.3. Thus E maps into r(G) XI (Aut(G) X TOP(W)) = (£(G) x Aut(G)) x TOP(W). This induces two homomorphisms E -^ i{G) x Aut(G) and E -^ TOP(W). Consequently, the action of JE^ on G x W is diagonal. Note that uniqueness does not hold in (€(G) x Aut(G)) x TOP(W) for T x g . The different strict equivalence classes are in 1-1 correspondence with H^{Q; r(G)) = conjugacy classes of representations of Q into r(G). 4.5.7. In [54], yet another construction is given in the (S4) case. Instead of G x W being the uniformizing space, G/K x W is chosen, where A^ is a maximal compact subgroup of G. Consequently, G/K is 2i simply connected symmetric space. The typical fiber becomes the locally symmetric space F = F \G/K and singular fiber is the quotient of F by 2ifinitegroup of isometries. The space E has a finite covering E' (perhaps branched)

K.B. Lee and F. Raymond

672

where E' is F ^ {Q'\W) and Q' is the kernel of Q ^ Out(r). Since Out(r) is finite, Q' has finite index in Q and the finite group Q/Q' acts diagonally on F x {Q' \W). In [42], this is explained in more detail by showing that the uniformizing group TO?i^G,K){G/K X W) is isomorphic to Isom(G//^) x TOP(W). In this context, existence, uniqueness and rigidity are all shown to be valid. Typical fibers which are locally symmetric spaces have a great deal of geometric interest and the uniqueness and rigidity of this modified construction has significant appHcations. When modeling on G/K x W instead of G X W we shall call this the modified injective Seifert Construction of type (S4). 4.5.8. Now we stack the two constructions together to obtain a homomorphism F -^ TOPG(G X wy. 1

^F

Q

1

^G

Q

-^1

Y

-^1

1 —^ M(W, G) X Inn(G) —^ T O P G ( G X W) —^ Out(G) x TOP(W) - ^ 1 This completes the proof of Theorem 4.3.2.

D

4.6. Smooth case If W is a smooth manifold and Q acts on it smoothly so that p\Q^^ Diff(W), then the construction can be done smoothly. The universal group is now C(W, G) x (Aut(G) x Diff(W)) which is certainly the subgroup of weakly G-equivariant diffeomorphisms of G X W. The same proof as for M(W, G) works in proving H\Q\ C ( W , Z ( G ) ) ) = 0, / = 1, 2, in the case of G completely solvable. We use this fact by simply referring to the "smooth Seifert fiber space construction". If W is a complex manifold and Q acts holomorphically, then one can also ask whether the construction can be done holomorphically on C^ x W. There are two types of obstructions. It could happen that the necessary C^-bundle over W, while trivial as a smooth bundle, may not be trivial as a holomorphic bundle. A more serious matter is that //2((2; W(W, C^)), where W(W, C^) denotes holomorphic maps, does not necessarily vanish and so not every smooth realization has a holomorphic one. Even when the first difficulty does not arise, the latter one may still persist. The solution to these problems and the corresponding theory is carefully worked out in [13]. One particular feature of the extended theory is that the uniformizing space need no longer be a product G x W but may be any principal G-bundle over W. See [41] for Seifert fibered spaces modelled on principal bundles and 6.2 for an illustration of the holomorphic theory. Another general approach to Seifert fiber spaces is due to Holmann [21]. His procedure extends the classical fiber bundle methods.

Seifert manifolds

673

5. Applications 5.1. Introduction The Seifert Construction, which is a special embedding, 0\7T ^^ TO?G{G X W ) , of the group 7T into T O P G ( G X W) such that 7t acts properly on G x 1^, preserves some of the properties of both G and W on 0 {it) \ {G yi W). Furthermore, the action of TT on G x W "twists" the topology and geometry of G and W to create the orbit space 0(n) \ {G yi W) in the same way that the group structures of F and Q "twist" to create the group n. In other words, this algebraic twisting of n makes the geometric twisting of the "bundle with singularities" r\G^

e{n) \{G^W)^

Q\W,

where the homogeneous space F \ G is a typical fiber. In the several applications, we have selected to include here this features seems especially prominent. One of the important geometric problems that has motivated the development of Seifert fiberings is the construction of closed aspherical manifolds realizing Poincare duality groups 7t of the form l-^r^^n^^Q^^l. The Seifert Construction enables one to find explicit aspherical manifolds M{n) when Q acts on a contractible manifold W and F is a torsion free lattice in a Lie group. For a topological manifold, the homotopy classes of self-homotopy equivalences can be regarded as algebraic data. We shall show how the Seifert Construction can be used to lift finite subgroups of homotopy classes to an action on the manifold. The final illustration is a description of the classical 3-dimensional Seifert manifolds and how they fit into our scheme of general Seifert fiberings. This description will also be useful for the final section.

5.2. Existence of closed smooth K{n, \)-manifolds There are two difficult problems related to the title. They are: • Which groups can be the fundamental group of a closed aspherical manifold? and • Ifn is the fundamental group of an aspherical manifold, can we give an actual explicit construction of an aspherical manifold for the group it ? There are some general criteria for the first problem such as TT must be finitely presented, have finite cohomological dimension and satisfy Poincare duality in that dimension. The Seifert Construction gives answers to both questions for large classes of groups n. The idea is that i f l ^ - F ^ - T T ^ - Q — > l i s a torsion free extension where F is the fundamental group of a closed aspherical manifold and 2 is a group acting properly on a contractible manifold W with compact quotient, then n should be the fundamental group of a closed aspherical manifold. We have the following THEOREM 5.2.1. Let F bea cocompact special lattice in G {see Section 4.3) and p\Q—> TOP( W) be a properly discontinuous action on a contractible manifold W with compact

674

K.B. Lee and F. Raymond

quotient. If I -^ F ^ n -^ Q ^> \ is a torsion free extension of F by Q, then for any Seifert Construction O'.n ^^ T O P G ( G X W), (1) M(0(n)) = 0(7t)\(G X W) is a closed aspherical manifold if F is of type (SI), (S2) or (S3), (2) M(0(7t)) = 0(7t) \ {(G/K) X W), where K is a maximal compact subgroup ofG, is a closed aspherical manifold if F is of type (S4). PROOF. We know from Theorem 4.3.2 that for each extension \-^F^^7z-^Q-^\, there exists a homomorphism of n into TO?G{G X W) (resp., TO?{G,K){GIK X W) in the case of type (S4), see [42] for this notation). We need only to check that this homomorphism is injective. Suppose QQ is the kernel of (p x p:Q -> Out(G) x TOF(W). Then (go is finite since the Q action on W is properly discontinuous. Let l->F-^E^'Qo-^l be the pullback via Qo C Q. The group E is torsion free since n is assumed to be torsion free. The restriction 0\E defines an action of £" on G x W (resp., G/K x W). Since F is of finite index in £, no non-trivial element of E can fix G x W (resp., G/K x W). For if it did, then some power would be a non-trivial element of F which does not fix G (resp., G/K). Therefore, 0 is injective; 0{n) acts properly and freely since it is torsion free. D REMARK 5.2.2. (1) If W is a smooth contractible manifold and p : g ^ Diff(W), then the construction can be done smoothly and M(0{7t)) is smooth. (2) If p\ and p2 are rigidly related (i.e., there exists h e TOF{W) for which p2 = l^(h) o p\)) and F is characteristic in TT, then M(0\(7T:)) and M(02(7r)) are homeomorphic via a Seifert automorphism. Moreover, if we fix i. and p, then the constructed M(0(7t)) are all strictly equivalent. (3) When W = {p} is a. point (a 0-dimensional contractible manifold), then Q must be finite for Q to act properly discontinuously, and every p: Q -> TOF{{p}) is rigidly related. The closed aspherical manifolds constructed are infra-G-manifolds. Cf. Example 4.4.4. (4) One important application of these constructions is that they provide model aspherical manifolds with often strong geometric properties. If one wants to study the famous conjecture that two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic via the methods of controlled surgery, then the constructed aspherical Seifert manifolds are excellent model manifolds.

The above procedure can be extended for even more general extensions. As an example. 5.2.3. Let n be a torsion-free extension of a virtually poly-1J group F by Q, where Q acts on a contractible manifold W properly discontinuously with compact quotient. Then there exists a closed K(7t, I)-manifold. THEOREM

PROOF. A torsion-free virtually poly-Z group F has a unique maximal normal nilpotent subgroup A, which is called the discrete nilradical of F. See [3]. Then the quotient F/A is virtually free Abelian of finite rank. Furthermore, since Z\ is a characteristic subgroup of r , it is normal in n. Consider the commuting diagram with exact rows and columns:

Seifert manifolds

i 1

675

1 I

-^ A ^

-^ r -

K

\-

1

1

1

1

Since F/A is virtually free Abelian of finite rank (say of s), it contains a characteristic subgroup 1/. Let Q' = {it/A)/If. Then the natural projection Q' ^^ Q has a finite kernel. Therefore, if we let Q' act on W via Q, the action will still be properly discontinuous. One can do a Seifert fiber space construction with the exact sequence 1

n/A -^

e'

1

which yields a properly discontinuous action of n/AonWyiW with compact quotient. Using this action oiTt/A onW x W, one does a Seifert fiber space construction with the exact sequence 1

7X

7t/A

1.

This gives rise to a properly discontinuous action of TT on A^ x (R^ x W), where A^ is the unique simply connected nilpotent Lie group containing Z\ as a lattice, with compact quotient. If the space W is smooth, and the action of g on W is smooth, both constructions can be done smoothly so that the properly discontinuous action of TT on A^ x (R^ x W) is smooth. In any case, since the group n is torsion free, the resulting action of TT on A/^ x (R^ x W) is free. Consequently, we get a closed K{7t, l)-manifold M = TT \ (A^xR' X W). It has a Seifert fiber structure

where the typical fiber F itself has a Seifert fiber structure A\N

—^ F —^ T' =Z'

\W.

In fact, since the action of the characteristic subgroup If on R^ is free, F is a genuine fiber bundle, with fiber a nilmanifold A\N over the base torus T^. D

676

K.B. Lee and F. Raymond

The space W does not have to be aspherical. As far as the action of Q is properly discontinuous, the construction works. The resulting action of ix is free if and only if the pre-image of Qy^ (the stabihzer of the Q action at each u; G W) in TT is torsion free. In this case, the space IT\{G yiW) will not be aspherical. See 4.3.2. 5.3. When is 6 injective ? Let 1 i

^ / ^ i

^7r \e

^Q

^1

Uxp ' 1 ^ M ( W , G) X Inn(G) — ^ T O P G ( G X W) —^Out(G) x TOP(W) —^ 1 be a Seifert construction. The argument as mentioned just above also shows that the action of TT on a general G x W will be free (and hence 0 is injective) if the pre-image of Qw (the stabilizer of the Q action at each u; G W) in TT is torsion free. Of course 0 may be injective, that is, IX acts effectively on G x W, without necessarily acting freely. We determine when the homomorphism 0\n -^ T O P G ( G X W) has trivial kernel for special lattices T in G of types (SI), (S2) or (S3) in Section 4.3.1. PROPOSITION

5.3.1. 0 is injective andO(n) O i{G) = 0(r) if and only if(p x p is injec-

tive. If (^ X p is injective, then clearly 0 is injective and ^(TT) Pi 1{G) = 0(r). Now assume that 0 is injective. Let K be the kernel of (p x p, and l^^F^^E^^K^^l the pullback via K C Q. Since K acts trivially on W, it is finite and E must map injectively into M(W, G) x Inn(G) = i{G) xziG) M(W, G). The group K maps injectively into M(W, G)/Z(G) by the natural projection i{G) ^z{G) M(W, G) -^ M(W, G)/Z{G), because we hypothesized that ^ (TT) fl t{G) =0(r). We complete the argument by showing M(W, G)/Z{G) has no non-trivial elements of finite order. This follows from the fact that the exponential map ^ ^- G is bijective for our special G. Therefore the equation x^ =a in G has a unique solution for any n G Z and every a G G. So if / G M(W, G) represents an element of K in M(W, G)IZ{G), then / ^ G Z ( G ) , where n is the order of / . Therefore, f^{w) =a,fox some a G Z(G), and all w eW. But there exists a unique x eG such that fiw) = X. Now, Z(G) is R^ for some k, and each a G Z(G) has a unique nth root in Z(G), X G Z(G). Since / has order n, « must be 1 and F = E. D PROOF.

5.4. Rigidity of infra-homogeneous spaces Recall from 2.9 the following definitions. Let G be a connected Lie group. A quotient of G by a lattice F is called a homogeneous space. More generally, let Aff(G) = G x Aut(G) act on G by (a,a)'X

=a •Qf(x).

Seifert manifolds

677

An infra-homogeneous space is a quotient of G by a group n C Aff(G), acting properly discontinuously and freely, such that F = n 0 G is a. uniform lattice of G and F has a finite index in n. Therefore, an infra-homogeneous space is finitely covered by a homogeneous space. For example, a flat Riemannian manifold may be called an "infra-torus". To emphasize G, sometimes we use the term infra-G-manifold for an infra-homogeneous space. Now we explain a little more detail for the claims made in Example 4.4.4. Let Z\/ be a special lattice of type (SI), (S2) or (S3). That is, 1 —^ ^/ ^

7ti ^ Q i ^ l

is an extension, Ot: Tit -> Aff(G/), Qi finite, pi: Qi -^ Out(G/) is injective, and hence by Proposition 5.3.1, Z\/ = 0i(7ti) H l{Gi), for / = 1,2. Suppose /i: TTi -> 7r2 is an isomorphism. Then, we claim, A\ is mapped isomorphically onto A2 and hence Q\ onto Q2. This isomorphism extends to an isomorphism of G\ onto G2. Thus we may think of Oi-.n-^ Aff(G) = TOPG ( G X {p}) such that Oiin) Pi G = Z\/, / = 1, 2. Now because p/: Qi -^ {p} is trivial, the embeddings are conjugate by an element of Aff(G). It remains to verify that h maps A\ isomorphically onto A2. It is well known that Z\/ is characteristic in the Abelian and nilpotent cases. For a proof of Z\/ being characteristic in the completely solvable case, we refer the reader to [35, Proposition 4.5]. Therefore h maps Z\ 1 isomorphically onto A2. Instead of appealing to the rigidity theorems for the Seifert Construction as we did above, we shall instead obtain these same extensions of the classical Bieberbach theorems directly. In the next theorem, the calculations involved resemble the proof of uniqueness of the Seifert Construction when W is a point, and show how closely linked the Seifert Construction is to the Bieberbach theorems. THEOREM 5.4.1. Suppose G has ULIEP {see Section 4.3.1), and H\^;G) (nonAbehan cohomology) vanishes for every finite subgroup ^ C Aut(G). Let TT, TT' G Aff(G) be finite extensions of lattices in G. Then every isomorphism 0 :n —> n' is a conjugation by an element ofAff(G). PROOF. Let F = G nn, F' = G HK' be the pure translations. Lei A = F n 0~\F'). Then ^U : A -^ 0{A) is an isomorphism of lattices of G. By ULIEP, 0\A extends uniquely to an automorphism C:G -^ G. Thus, 0\A = C\A, and hence, 0(z, I) = (Cz, /) for all (z, I) £ A. Let us denote the composite homomorphism

7t-^7i'^

Aff(G) -^ Aut(G)

by (9, where Aff(G) = G xi Aut(G) -^ Aut(G) is the projection. Let ^ = n/A. Then clearly, 0 factors through ^. Define a map A: TT ^- G by 0(w, K) = {Cw • X(w, K), 0(w, K)).

(1)

678

K.B. Lee and F. Raymond

It is easy to see that X{zw, K) = X{w, K) for dXl z ^ A. Therefore, X is really a map

For any (z, I) e A and (u;, K) e n, apply 0 to both sides of (w, K)(z, I)(w, K)~^ = (w ' Kz ' w~\ I) to get Cw ' X(w, K). e{w, K)(Cz)' A.(u;, K)'^ - (Cw)-^ =0{w • Kz • w'^). However, w - Kz - w~^ e A since A is normal in n, and the right-hand side equals to C(w ' Kz ' w~^) = Cw ' CKz • (Cw)~^ since C: G ^- G is an automorphism. From this, we have 0(w, K)(Cz) = Hw, K)-^ • CKz • A(M;, K).

(2)

This is true for all z G T. Note that 0(w, K), C and K are automorphisms of the Lie group G. By ULIEP of G, //i^ equality (2) /zo/J5' true for all z^G. We claim that, with the ^-structure on G VmO:^ ^ Aut(G), A G Z\^\ G ) ; i.e., A,: »/^ ^^ G is a crossed homomorphism. We shall show X((w;, K) • (u;', /i:')) = A(u;, ii:) • ^(u;, /r)(A(w;', K')) for all (u;, A'), (w;^ K') G TT. (Note that we are using the elements of n to denote the elements of ^.) Apply 0 to both sides of {w,K){w\K') = {w • Kw\KK') to get Cw ' Hw, K)' 0(w, K)[Cw' • X(w\ K')] = C{w • Kw') • X((w, K){w\ K')). From this, it follows that X{{w, K)(w\ K')) = (CKwY^

' Hw, K) • e{w, K){Cw') • e{w, K)X(w\

K').

By (2), we have e(w,K){Cw') = X(w,K)-^ • CKw' • X,iw,K) so that X((w,K) • (u;^ K^)) = k(w, K) • 0(w, K)X(w\ K'). This shows that A, is a crossed homomorphism. By the assumption, H\^; G) vanishes. This means that any crossed homomorphism is "principal". In other words, there exists d eG such that X(w,K)=d'0(w,K){d~^).

(3)

Let D = iJi(d~^) o C. We check that 0 is the conjugation by (d, D) = (d, /x(J~^) o C) G Aff(G). Using (1), (2) and (3), one can show 0(w, K) o /x(J-^) o C = MC^"^) OCOK. Thus, for any (u;, K) G n, 0(w,K)'(d,D) = {Cw • X(w, K), 0(w, K)) • (J, ix{d-^) o C) = {Cw • Xiw, K) • ^(it;, K){d), 0(w, K) o /x(t/-*) o C) = (Cu;. J • ^(u;, /r)(j-^) • Oiw, K)(d), Oiw, K) o /x(^-^) o C)

Seifert manifolds

= =

679

{Cw'd,fi{d-^)oCoK) (d,D)'(w,K).

This finishes the proof of our claim.

D

COROLLARY 5.4.2. Suppose G has ULIEP, and H\^; G) (non-Abehan cohomology) vanishes for every finite subgroup ^ C Aut(G). Then homotopy equivalent infra-Gmanifolds are affinely diffeomorphic.

Let M = TT \ G, M' = TT' \ G' ht infra-G-manifolds. A homotopy equivalence M -^ M' induces an isomorphism O'.TT -^ 7^^ Since 0 maps some subgroup of GHTT into a lattice of G\ there is an isomorphism of G onto G^ Using this isomorphism, we identify G with G^ Now apply Theorem 5.4.1 to 0\TC -^ TC' to find an element h = (d, D) e Aff(G) which conjugates n onto 7T\ This gives a weakly equivariant map PROOF.

(/i,M(/i)):(G,7r)^(G,7rO and h gives rise to an affine diffeomorphism M -^ M\ which is homotopic to the original map. D COROLLARY 5.4.3. (1) (Bieberbach) Homotopy equivalent flat manifolds are affinely diffeomorphic. (2) [39] Homotopy equivalent infra-nilmanifolds are affinely diffeomorphic. (3) Homotopy equivalent infra-solvmanifolds of type (R) are affinely diffeomorphic.

Notice, in particular, that the isomorphism 6 maps the lattice F = G On onto F^ = This is also true topologically on the orbifold level.

GHTT^

PROOF. What is needed for this conclusion is the element h = (d, D) e Aff(G) which conjugates 7t onto 7^^ We have this element immediately from the rigidity of the Seifert Construction for lattices of type (SI), (S2) and (S3). If we wish to avoid the Seifert Construction, then we need to verify that H^(^; G) vanishes for each finite subgroup ^ C Aut(G). This is accomplished in [39] and [35], and well known for G of type (SI). D

5.5. Lifting problem for homotopy classes of self-homotopy equivalences Let M be an admissible space (see Section 2.5) and £(M) be the //-space of homotopy equivalences of M into itself. Any / e SiM) induces an isomorphism /* : TTI (M, X) ^^ 7t\(M, f{x)). By choosing a path co from x to /(JC), we have an automorphism f^ of 7Z\ (M, x), defined by /jf ([r]) = [(j)~^ • ( / o T^) • ^ ] - A different choice ofco alters f^ only by an inner automorphism. Therefore, we obtain a homomorphism y'.£{M)^0\xi{7z),

680

K.B. Lee and F. Raymond

7t = 7T\ (M, x).lf M is 2i K(7T,l) space, then So(M) is the kernel of y so that y factors through no(£(M)) = £(M)/£o(M), where £o(M) is the self homotopy equivalences homotopic to the identity. Moreover, y is onto since every automorphism of n can be realized by a self homotopy equivalence of M. A homomorphism cp: F ^^ Out(7r) = 7to(£(M)) is called an abstract kernel. An injective abstract kernel is the same as a subgroup of homotopy classes of self-homotopy equivalences of M. A lifting of <^ as a group of homeomorphisms is a homomorphism (p'.F -^ TOP(M) which makes F

.^

V TOP(M) — ^ £(M)

F (p

— ^ 7to(£(M))

commutative. Suppose F acts on M. Let F* be the group of all liftings of elements of F to the universal covering M. Then 1 ^ TT ^ F* -^ F ^ 1 is exact, and is called the lifting sequence of the action (F, M). Furthermore, for an aspherical M, if (F, M) is effective, then the centralizer ofn in F*, CF*(7r), is torsion-free (such an extension was called admissible in [36]); and F* lies in TOP(M). Thus we have a necessary condition for the existence of a lifting of an abstract kernel (p: F -> Out(7r), as an (effective, resp.) group action: the existence of an (admissible, resp.) group extension of TT by F realizing the abstract kernel. For finite groups, this necessary condition is also sufficient for some tractable manifolds. No examples of closed aspherical manifolds are yet known where the existence of a group extension does not also yield a group action. 5.5.1. Let Q act properly on an admissible space W (see Section 2.5) and B be the quotient Q\W. Suppose for each extension l^^Q-^E^^F^^lbya. finite group F, the action of Q extends to a proper action of F on W. Then we say that the Q action on W is finitely extendable. In particular, then F acts on B preserving the orbit structure. If r is normal in TT, let Aut(7r, F) denote the automorphisms of TT that leave F invariant. Since Inn(7r) leaves F invariant, we can put Aut(7r, F)/Inn(7r) = Out(7r, F ) . It is a subgroup of Out(7r). See notation in Section 1. Let M = TT \ M be an orbifold (with the orbifold group n). Call a finite abstract kernel F -> Out (TT) geometrically realizable if it can be realized as an action of F on M.

DEFINITION

We are interested in realizing a finite abstract kernel F -^ Out (TT) as a group action on a Seifert fiber space M with a typical fiber F \ G. In particular, we want the F action to be fiber-preserving maps. This means that, on the group level, the extension must leave

Seifert manifolds

681

the lattice F invariant. In other words, we consider only those abstract kernels which have images in Out(7r, F). 5.5.2. Let F be a lattice in the special Lie group G (see Section 4.3.1), and p: Q ^^ TOP(W) be a proper action on W which is finitely extendable. Let \ -^ F ^^ 7T ^^ Q ^^ I be an extension and O'.n -^ T O P G ( G X W) be any homomorphism. Then each abstract kernel (p: F -^ Out(7r, F) of a finite group F can be geometrically realized as a group of Seifert automorphisms on M{0(7T)) = 0{n) \ (G x W) if and only if the abstract kernel (f admits some extension. THEOREM

PROOF. Let 1 ^

TT ^

£" ^

F ^- 1 be an extension realizing the abstract kernel (p. Con-

sider the induced extension 1

Q -^

E/F

1.

Since p is finitely extendable, there exists p^: E/F -> TOF(W) extending p:Q ^^ TOF(W). We have the commutative diagram with exact columns and rows: 1

-^ F

1

-^ 1

7T

E/F ^

-^ F

1

-^ F 1

1

Again by the existence theorem for special lattices, Theorem 4.3.2, there exists 0' :E ^^ T O P G ( G X W), where 6>V = 6>|r = /: T - ^ G, and P'\Q = p. Put(9'U = 6>^ Of course, 0' may be different from 0, but as 0 and 0' agree on F and Q, we can apply Theorem 4.3.2(2) to conjugate T O P G ( G X W) by an element of M(W, G) x Inn(G) which carries 0^\jj: to 0 so that the new homomorphism 0' '.E ^^ T O P G ( G X W) is an extension of ^: TT -^ T O P G ( G X W). This yields an action of F on 0{7r) \ (G x W) as a group of Seifert automorphism as desired. D Since we used the uniqueness of the Seifert Construction, we use the modified Seifert Construction modelled on G/K x W instead of G x \y in the semi-simple case. COROLLARY 5.5.3. Let M(7t) be an infra-G-manifold with G special, and with fundamental group n. Suppose cp: F ^^ Out(7r) is an abstract kernel. Then there exists a geo-

682

K.B. Lee and F. Raymond

metric realization of F acting on M(7t) by affine dijfeomorphisms if and only if there is an extension EofnbyF which realizes this abstract kernel. PROOF. The infra-G-manifold M{7T) is modelled on G x {/?} {p = point) {G/K x [p] for a convenient form of G in the semi-simple case) and T O P G ( G X {/?}) = Aff(G) (resp. Aff(G, K)), see [42] for this notation. Trivially, every Q -^ TOP({p}) extends to E/T -^ T0P({/7}). The above theorem then immediately applies, and F acts on M(7r) by Seifert automorphisms which are affine diffeomorphisms. D REMARK 5.5.4. (1) Since we may introduce a metric structure in the corollary from a left invariant metric on G (see Section 2.9), M(7T) has the structure of a flat, almost flat, Riemannian infra-solvmanifold or a locally symmetric spaces. We may also further conjugate 0(7t) in Aff(G) so that F now acts on the conjugated manifold by isometrics preserving the flat, etc., structures. (2) The proper action of n in the theorem is not necessarily free nor effective. Thus M(7T) could very well be a Seifert orbifold. The corollary then works for such orbifolds, i.e., infra-G-spaces. In the Euclidean case, M(7t) would then be a Euclidean "crystal" and 7t a Euchdean crystallographic group. In Theorem 5.5.2, F sends fibers to fibers. For more about the realizations up to strict equivalences, and finding examples where F does not lift because there are no extensions realizing the abstract kernels, the reader is referred to [22,26,32,33,36,37,42,53,59,64] and [52].

5.6. Seifert fiberings with codimension 2 fibers 5.6.1. There is a class of Seifert fiberings which are very close generalizations of the classical Seifert 3-manifolds. Let the discrete group Q act effectively and properly on the topological plane R^. Then Q can be conjugated in TOP(M^) to be a group acting as isometrics on the hyperbolic plane or the Euclidean plane. In the first case, call Q hyperbolic, and in the latter Euchdean. If Q preserves orientation, then Q can be conjugated to act as holomorphic actions on C or the unit disk D. This latter fact, using standard techniques in transformation group theory, uniformization and topology of 2-manifolds, is not very difficult to prove. (Certainly, if Q acts simplicially, the arguments are not hard.) When the orbit space g \ M^ is compact, then a theorem, essentially due to Nielsen, says that any two Q actions are topologically conjugate. If the actions are assumed smooth, then they are smoothly conjugate. In the non-compact case, there are similar results but they can be very complicated if Q is not finitely presentable. So for our next example, we restrict to Q for which (g \ R^ is compact. Since 2 \ R^ will have the structure of an orbifold, Q will either be isomorphic to a Euclidean crystallographic group or a cocompact hyperbolic group. In the Euclidean case, these are the 17 wall-paper groups. They are centerless except for l3 = ^ x Z o r Z x ) Z (the fundamental group of the Klein bottle). In the hyperbolic case, all the groups are not solvable, and have no non-trivial normal Abelian subgroups. 5.6.2. An important class of closed 3-manifolds are those closed 3-manifolds whose fundamental group contains a normal subgroup Z. This class contains all the classical Seifert 3-manifolds with infinite fundamental group as well as some other non-orientable manifolds such as closed 3-manifolds admitting a circle action without fixed points but having

Seifert manifolds

683

2-dimensional submanifolds whose stabilizer is Z2. The orientable ones, however, coincide with the classical orientable Seifert 3-manifolds with infinite fundamental group as defined by Seifert in [60]. Of course, these statements are to be taken modulo the Poincare Conjecture, for if there were a fake 3-sphere U, then MttZ', where M is a Seifert manifold, would not be a Seifert manifold but would still have a normal Z in its fundamental group. In any case, a closed 3-manifold whose fundamental group contains a normal Z subgroup, has, modulo the Poincare Conjecture, an orientable cover admitting an injective 5^-action. Consequently, they all have a universal cover that splits into R x R^ or R x 5^, where the R descends to the fibers associated with the normal Z. In the R x 5^ case, the manifolds have an orientable cover isomorphic to 5^ x 5^. Thus, their fundamental groups are just finite extensions of Z. We are interested in those that are not finite extensions of Z. They are all aspherical since they are covered by R^. Therefore, for each extension, I ^ - Z ^ - T T ^ - Q-> 1, where p:Q-^ TOP(R^) is an effective proper action on R^, we may construct model Seifert manifolds. In fact, as p is topologically rigid, we can assume that after conjugating in TOP(R^), p maps Q into either IsomCE^) = R^ >^ 0(2) or Isom(HI) = PSL(2, R) x Z2, the isometrics of the Euclidean plane or the hyperbolic plane. In this case, as we shall see in Section 6.3, the constructed Seifert manifolds possess a geometric structure inherited from a subgroup U of T O P M ( R X R 2 ) into which 7t\ (M) is embedded. The group n acts freely on R x R^ if and only if 71 is torsion free. Otherwise, the map R x R^ ^- ^(TT) \ (R x R^) is a non-trivially branched covering and the underlying topological space of the 3-dimensional orbifold, which is still a 3-manifold with possible boundary, may not even be aspherical (see the next section). To obtain all possible orbifolds, we have to enlarge Q to Q', where Q is of index 2 in Q' and the Z2 in Q' does not act effectively (it injects into Aut(Z)). That is, (^ X p : 2 ' -^ Aut(Z) X TOP(R) -^ Aut(R) x TOP(R^) will be injective. Obviously the existence of the Seifert Construction for every torsion free extension l ^ - Z - ^ T T ^ - Q - ^ l produces all the known aspherical 3-manifolds with a normal Z in their fundamental group. Furthermore, rigidity of the construction, since Z will almost always be characteristic, says that any two constructed manifolds with isomorphic fundamental groups are diffeomorphic. In fact, except for several exceptions (less than 10), the isomorphisms must be given by a Seifert automorphism. (The exceptions can occur for the 3-torus, for example, where the Z of the fiber is not the only possible normal Z.) That no other manifolds are possible except those given by the Seifert Construction follows from results of Waldhausen in the Haken case. For non-Haken manifolds, the result is more recent and is due to Scott, Casson and others. The rigidity result is also true in the orbifold case. For example, if Q is orientationpreserving and Q -^ Aut(Z) is trivial, then the R action descends to an 5^-action on ^(TT) \ (R X R^). The group Z is the maximal normal Abelian subgroup of n in case Q is hyperbohc since Q has no normal Abelian subgroups. In the Euclidean crystallographic case where Q ^1?, Q is centerless and so Z will be the entire center of n and so is characteristic. For Q = Z^, the TT are torsion free and are distinguished by H\ (TT; Z) = Z^ + Z/,. So, except for 3-torus, the rigidity theorem for the Seifert Construction implies two such oriented orbifolds with isomorphic orbifold groups will be diffeomorphic via an S -equivariant orientation-preserving homeomorphism.

684

K.B. Lee and F. Raymond

THEOREM 5.6.3 (Higher-dimensionalfibers).Let Q be hyperbolic and act properly on M^ with compact quotient. Let F be a special lattice of type (SI), (S2) or (S3) {see Section 4.3.1), and

l^r-^TT—> Q ^

1

be an extension. Fhen there exists a homomorphism 0 :n ^ T O P G ( G X R ^ ) unique up to conjugation, TT acts freely if and only ifn is torsion free and, consequently, M(0(7t)) will then be aspherical. In general, M(0\ (TTI)) is homeomorphic via a Seifert automorphism to M(02(7t2)) if and only ifni is isomorphic to 7T2PROOF. We know 0 exists by the existence theorem for special lattices. Since Q has no non-trivial normal solvable subgroups, and G is solvable, 7t HG = F. Therefore, F is the maximal normal solvable subgroup in n and consequently is characteristic. Since p: Q -^ TOP(R^) is rigid, the Seifert Construction is rigid and the theorem follows. (Cf. [13, Section 12].) D

Nicas and Stark in [47], using controlled surgery, have shown that any closed aspherical topological manifold whose fundamental group is a central extension of Z^ by a 2dimensional orientation preserving cocompact hyperbolic Q is homeomorphic to the corresponding Seifert manifold constructed in Theorem 5.6.3, provided the dimension of the manifold is greater than 4. COROLLARY 5.6.4. Let (p: F -> Out(7r) be an abstract kernel, where F is finite and n is as in the above theorem. Then (p has a geometric realization of an action of F as a group of Seifert automorphism on M{0{TC)) if and only if there exists some extension Eofn which realizes the abstract kernel.

The extension E induces a short exact sequence 1 ^ - Q ^^ E/F ^^ F ^^ 1. S. Kerckhoff [23] has shown, in his solution to the Nielsen Realization Problem, that a finite extension of a hyperboHc Q with an injective abstract kernel is again a hyperboHc group. If K denotes the kernel of the abstract kernel F -^ Out(2), then (E/r)/K is a hyperbolic group containing a topological conjugate, in TOP(R^), of Q. Then E/F, via E/F -^ (E/F)/K, acts properly on HI and (up to conjugation in TOP(R^)) extends the Q action. Therefore Theorem 5.5.2 applies. D PROOF.

5.7. The classical Seifert 3-dimensional manifolds In a classical paper [60] in 1933, H. Seifert described a class of 3-dimensional manifolds that have turned out to be fundamental for the topology of 3-manifolds. We assume the reader has some familiarity with these manifolds. In the closed orientable (resp. non-orientable) case Seifert's manifolds whose fundamental groups are infinite, but not a finite extension of Z, coincide with (resp. are a subset of) those closed orientable (resp. non-orientable) manifolds, ST, that can be constructed, as

Seifert manifolds

685

in Proposition 4.3.2, from a torsion free extension, \^^Z^^7Z^^Q-^\, with Q acting properly, effectively, and cocompactly on R^. It has recently been shown that if there are no fake homotopy 3-spheres, then the closed 3-manifolds, whose fundamental groups contain a normal Z subgroup and which are not finite extensions of Z, coincide with SJ^. See 5.6.2. 6. Reduction of the universal group 6.1. Purpose and requirements In a general Seifert Construction, it may happen that p : 2 —> TOP(W) has an image in a subgroup that has geometric or topological significance. This means that it is likely that the associated Seifert Constructions inherit some of these properties. Let ZY be a subgroup of T O P G ( G X W) containing £(G). Put z7 = ZY H (M(W, G) x Inn(G)) mdU = U/Uso that 1

^U

^U

n 1 - ^ M(W, G) X Inn(G) -^

n T O P G ( G X W) ^

^U

^1

n Out(G) x TOP(W) -^

1

(6.1) commutes. If the image of ^ : TT ^- T O P G ( G X W) lies in U, we say that the universal group has been reduced to U fox n. The group hi can then be used to study extra structures on the Seifert fiber space Gin) \ (G x W). We have already seen in Section 4.4 that if W is a smooth manifold, F special and p'.Q^^ Diff(W), then any injective Seifert Construction can be done smoothly in DiffG(G X W)=C{W, G) X (Aut(G) x Diff(W)), where DiffG(G x W) = T O P G ( G X W ) H Diff(G x W). Furthermore, existence, uniqueness and rigidity also hold because we may use a differential partition of unity in the proof for the vanishing of the necessary cohomology groups. To obtain existence of an injective Seifert Construction in hi, one has to verify the conditions in Theorem 3.7.2 in hi. If hi is much smaller than T O P G ( G X W), uniqueness and rigidity are not likely to hold. The set of all homomorphisms of n into hi which belong to the same conjugacy class in T O P G ( G X W ) is a deformation space of that conjugacy class. We shall illustrate these concepts by describing in some detail the deformation spaces for interesting geometric situations. 6.2. Injective holomorphic Seifert fiberings We assume that G = C^, W is a complex manifold, p:Q -^ TO?{W) has image in the holomorphic homeomorphisms of W, and H{W, C^) C M(W, C^) are the holomorphic maps.

686

K.B. Lee and F. Raymond

We also assume that the map C^ x 1^ ^- W is holomorphically trivial and so U = Hol(^k (C^ X W) then becomes n{W, &) X (GL(i^, C^) X Hol(W)), where Hol(\y) is the group of holomorphic automorphisms of W. Existence, uniqueness and rigidity do not necessarily hold because we do not have holomorphic partitions of unity and the groups H^ (Q; H(W, C^)) does not vanish in general. The reader is referred to [13] where a general and comprehensive theory of holomorphic Seifert fiberings whose universal space is a holomorphic fiber bundle over W with fiber a complex torus or C^ is given. We shall restrict ourselves here to a special case closely related to the classical Seifert 3-manifolds. Let (C*, M) be an injective, proper, holomorphic C* action on a complex 2-manifold M so that the quotient space is compact. As in the case of an injective S^ action, an injective proper C* action lifts to the covering space of M corresponding to the image of the evaluation homomorphism, and yields a splitting (C*, C* x W) where W is a simply connected complex 1-manifold (cf. Section 2.2). Therefore, W is C, D the open unit disk, or CP{. We shall restrict ourselves to W being the unit disk D. The orbit space Q\W is a. closed Riemann surface. The action of Q = 7T\ ( M ) / Z on D is holomorphic, properly discontinuous, but not necessarily free. Therefore, M ^ - C * \ M = ( 2 \ W i s a generalization of a principal holomorphic C*-bundle over a Riemann surface. From the exact sequence 1 ^- Z -> C —> C* -^ 0, we obtain the exact sequence

0 - ^ z = M(w, z) - ^ n(w, C) - ^ mw, c*) -^ 0 which gives rise to a long exact sequence of cohomology groups . •. ^

W(Q; Z) -^

W{Q- n{W, O ) -^

W{Q; H(W, C*)) ^

• • •.

The group Q acts on the unit disk Z) as a cocompact Fuchsian group. That is, p: Q -^ Hol(D), the complex automorphisms of the unit disk. The action of (2 on A G HiW, C) is given by "A = A o ( p ( a ) ) " ^ Let us compare the smooth situation with the holomorphic one. We have the following commutative diagram of exact sequences

— ^ / / ' ( 2 , Z)^H\Q,

C(W, O ) — ^ H \ Q , C{W,

1^

C * ) ) ^ / / 2 ( e , Z)—^//2(g. ^(^^ Q )

(6.2) For the smooth case, H'{Q; C{W, C)) = 0, j" > 0, and as we shall see, H^{Q; n{W, C)) = 0.

Seifert manifolds

687

For each central extension Q -^ Z -> n ^^ 2 ^ ^ 0 represented by [/] G H^(Q\Z), we have smooth Seifert Constructions O'.n ^^ Diffc(C x D) = Diffc(C x R^). If we fix /: Z ^- C and p: Q ^^ Diff(D), the construction is unique up to strict equivalences (Section 4.4.2). We have the smooth Seifert orbifold over Q\W = Q\D with an induced C* action and therefore an S^ action on 0(7t) \ (C x D) (= R^ x 0(7t) \ (R^ x D) = R^ x A^^ since C* splits smoothly as R^ x S^). The uniqueness says that for any other embedding 0' \n -^ Diffc(C X D), keeping / and p fixed, the C* action on ^^(TT) \ (C x D) is strictly smoothly equivalent to that on ^(TT) \ (C x D). For the same n, we have the homomorphism H^{Q\ Z) -> H^{Q\ H(D, C)). The second group fortunately can be identified with the second cohomology of the sheaf of germs of holomorphic functions over Q\ D. This vanishes since Q\ D is (complex) 1-dimensional and the sheaf is coherent (i.e., locally free). This means that [/] € //^(Q; Z) maps to 0 G H^(Q; H(W, C)). But as the groups are Abelian, this becomes exactly the identity (4.1), and we have 0 \7i ^^ Holc(C x W). Therefore, each [/] has holomorphic realizations for each fixed / and p. Recall from Theorem 3.7.2, the set of all ^ : TT ^- Holc(C x D) with fixed /: Z -> C and p'.Q^^ 'H{D), up to conjugation by elements of 7Y(D, C), is in one-one correspondence with H^(Q;H(D, C)). (This complex vector space is the same as H\V;h^), the first cohomology of the sheaf of germs of holomorphic functions where V is treated as the analytic space V = Q\ D. That is, for each open f/ in V, we consider p~\U) and holomorphic functions X: p~\U) -^ C such that X(p(a)(w)) = X(w), w € p~^{U) and p:D ^^ 2 \ ^ is the projection. This defines the sheaf h^ over V.) This group is isomorphic to C^, where g is the genus of V. 6.2.1 ([13, §13]). For each smooth action (C*,M) corresponding to the unique strict conjugacy class 0(7t), with it torsion free, there exists a complex g-dimensional family of strictly holomorphically inequivalent C* actions each strictly smoothly equivalent to the smooth (C*, M). THEOREM

PROOF. We may interpret [/'] G H\Q\ H{W, C*)) to represent the effective holomorphic C* action on ^(TT) \ (C x D) up to strict C* equivalence. If Q were torsion free, then Q acts freely and 2 \ D = V is a closed oriented surface without branch points. The C* action is then free and proper yielding a principal holomorphic C* bundle, corresponding to a complex line bundle over V. Since Q is not assumed to be torsion free, f determines the holomorphic C* action (cf. [13, §5]). Given two injective C* actions [/^] and [/''] with the same "Chem class" 6{f'] = 8[f% there exists 2i[X]eH\Q; n(D, C)) such that [/^^] = [/^] + [X]. Since H^{Q\ H(D, C)) is a vector space of complex dimension g, there exists a whole g-dimensional family of inequivalent holomorphic C* actions starting from [/'] and ending with [f^]. D Returning to diagram (6.2), define Fic(Q\D)

= //i(e;7Y(Z),C))/image(i/He;Z)) = a complex g-torus, or real 2g-torus, T^^.

688

K.B. Lee and F. Raymond

The connected component of the group H\Q; TL{D, C*)), in the Q torsion free case, is called the Picard group for the line bundles over Q\D.\n our case, H\Q\ H(D, C*)) is isomorphic to T^^ 0 Z 0 finite torsion. We obtain the exact sequence 0 -^

Pic(e \ D) —> //^ ( 2 ; W(D, C*)) -^

H^(Q; Z) -^

0,

where the middle group is the isomorphism classes of injective holomorphic C* actions over Q\D,8 sends such an isomorphism class to its "Chem class" and Pic(2 \ D) represents the deformations. As before, H^(Q;Z) = Z^ Torsion. In the discussion above, we have fixed / :Z ^- C and p: Q ^^ Hol(D). If we vary these choices, we don't get anything new in the smooth case because of smooth rigidity. That is, 0(7T) is conjugate to 0^(7t) in Diffc(C x D) where conjugation is taken in the whole group and not just in C(D, C) as for strict equivalence. However, in the holomorphic case, a change in p: Q ^^ Hol(D) induces a much larger deformation space than treated above. We can see this in our next example of reduction of the universal group where instead of considering complex structures and complex actions, we replace them by essentially equivalent Riemannian metric structures and metric preserving 5'-actions on N^ (M = N xR^).

6.3.

FSL(2,R)-geometry

6.3.1. When W is homeomorphic to R^ in a Seifert Construction and p{Q) is a. discrete subgroup of TOP(R^), acting properly on R^, then the group p(Q) can be conjugated in TOP(R^) to a group which acts as isometrics on R^ with the usual Euclidean metric (e.g., p(Q) is crystallographic) or as isometrics on R^ with the usual hyperbolic metric. In the former case, we say p(Q) is isomorphic to a (Euclidean) crystallographic group, and in the latter, p(Q) is isomorphic to a hyperbolic group. We write R^ with the usual Euclidean metric (resp. hyperbolic metric) as E^ (resp. H). If 2 \ R^ is compact, then any two embeddings of Q are conjugate in TOP(R^). However, if p\, pi'-Q ^^ Isom(HI) have compact quotients, then p\ is conjugated in Isom(EI) to p2 if and only if P2(Q) lies in the normalizer of p\(Q) in Isom(EI) (the normalizer is a finite extension of p\(Q)). Thus, if we reduce the universal group T O P G ( G X R ^ ) to U, where at least TOP(R^) is replaced by Isom(E^) or Isom(H[), we would expect to find a rich deformation theory for Seifert Constructions modelled on G x R^. We will take G = R, and Z7 = Isom(HI) c T 0 P ( R 2 ) . For each central extension O - ^ Z ^ - T T ^ - 2 ^ - 0 with Q cocompact hyperbolic, the group n can be embedded by ^ ::T ^- TOP]R(R X M) SO that it is topologically and/or smoothly rigid. If n is torsion free, TT \ (R x H) is a classical closed Seifert 3-manifold N^ and with a unique S^ action up to equivalence. The 5'-orbit space or base space is a 2-dimensional orbifold isomorphic to Q\M.. The product R x EI carries several geometries so that the Riemannian metric induced on R x HI ^- IH is the hyperbolic metric. We will examine first the Riemannian metric on PSL(2, R) and then later an R-invariant Lorentz metric on R x IHI.

Seifert manifolds

689

Let us denote PSL(2, R) by P. The space P is the universal covering group of PSL(2, R). Topologically P is homeomorphic to R x H. P = PSL(2,R), P = PSL(2, R)

Therefore, if we use TOPM(R X H ) = M(H, R) X (GL(1, R) x TOP(H)) as our universal groug, we will not be able to distinguish the geometries of R x H, R x R^ and Nil from that of P. The Lie group PSL(2, R) can be viewed as the unit tangent bundle of the hyperbolic space H, and it has a natural Riemannian metric. This metric pulls back to a Riemannian metric on P. It turns out that this metric is right invariant. That is, all right translations by elements of the group are isometrics. Furthermore, the isometry group is Isom(P) = (R x z P) x Z2, where R is a subgroup of P containing the center Z, acting as left translations. These two actions commute with each other, and i{z) = r(z~^) for z G Z, the center of P. The finite group Z2 is generated by the reflection about the j-axis. In fact, any orientation-reversing isometry of period 2 will do. While it reverses the orientation of the base space H, it also reverses the orientation of the fiber R. Consequently, it preserves the orientation of P. We take the subgroup R described above as our G. Smoothly, P = R x H. Therefore, we have G = R,

W = m.

Since R xz P commutes with the left translation G = R, and the generator of Z2 is an inversion of R, Isom(P) = (R x z P) x Z2 lies inside TOPM(R^X H ) . TO make the presentation clearer, we use only the connected component of Isom(P). So, let's take U = lsomo(P) =R xz P so that we have the commuting diagram: 1

^ R

n

^U = RxzP

n

^ P

^ 1

n

1 — ^ M(H, R) — ^ TOPM(R X H) — ^ GL(1, R) x TOP(H) — ^ 1

For this case, Q coincides with the Q examined in Section 6.2, and has a well known presentation

690

K.B. Lee and F. Raymond

\

j=\

i=\

I

for p ^ 0, g ^ 0 and all aj > 2. It is also required that the Euler characteristic of Q, defined as X(Q) =

(2-2g)

U-B

satisfies x(Q) < 0- It is our intention to characterize those [TT] G H^(Q; Z ) which embed in U and to determine their deformation spaces. 6.3.2. For any subgroup Q of Isomo(HI), one can pullback (see Subsection 3.3.2) the above extension via Q ^^ Isomo(IHI) to get Q so that the diagram 1

-^ 1

^Q

Q

Isomo(H)

Isomo(P)

commutes. Thus, M becomes a (trivial) Q-module. 6.3.3. Let Q be a cocompact discrete subgroup o/Isomo(lHI). Then (1) //2(G;M):^R. (2) The class [Q\ e H^{Q\ M) is non-zero. {That is, the exact sequence does not split.)

LEMMA

(1) Let 1 ^- Z ^- P ^- P -> 1 be the universal covering projection, and let 1 ^• Z ^- 2 ^^ 2 ^^ 1 be the pullback of this exact sequence v'm Q^-> P. Then 2,(1 Q sits in M x^ P = IsomoP as the center. Denote the inclusion of Z ^^ R C Isomo(P) by /. Then PROOF.

1- ^ Z ^ i

' 1—^li ^ i

Q ^

Q--^ 1

\ V

l ^ Q - -^ 1

{Q]eH 2(G;Z) ' /* ' [Q]eH H.Q;M.)

is commutative so that i^[Q] = [ 2 ] . By Selberg's lemma, Q contains a torsion free normal subgroup So of finite index. Then H^{QQ\W) = H^{Qo\ H; R) = R. Then, by transfer, //2(|3; M) = R, since Q/QQ acts on 2o \ M preserving orientation. Since H^{Q\ Z) is finitely generated, it follows that //^((g; Z) = Z 0 Torsion by the Universal Coefficient Theorem, and the fact that i^:H^{Q\Z)-^ H^{Q\ R) is given by 0 R. Thus the elements of infinite order inject and those of finite order are in the kernel. (2) We can assume, without loss of generality, 2 C P is torsion free. In [55], it is shown that {Q\ G H'^{Q\IJ) is non-zero. In fact, Q is the fundamental group of the unit tangent

Seifert manifolds

691

bundle Q\P of the surface Q \ H. In this case, the Euler characteristic of Q,2 — 2g^0,is the characteristic class of the principal 5 ^-bundle Q\P -^ 2 \ H, and is also equal to the negative of the cohomology class [iQ] e //^(Q; Z) = Z of the extension Q. Therefore, [Q] is non-zero in H^(Q;R) = H^(Q;Z) 0R. We also point out, in this case, [Q] = e(Q), see Subsection 6.3.4 for definition of e{Q). D 6.3.4. (Euler number) Let /: Z ^^ M be the standard inclusion. For each central extension 0 ^ - Z - ^ 7 r - > <2-> 1, there is associated a rational invariant called the Euler number of TT, and is denoted by ^(TT). It can be defined in terms of a presentation of TT. Let 7t = I x\, ...,Xg,y\,...,yg,

w

j=\

w\,...,Wp,z\z

i=\

ctniral,

I

Then e(n) = —{b -\- E ^) and |^(TT) | is an invariant of the isomorphism class of TT . In [31, Theorem 4.5], it is shown that under the homomorphism /*: H^{Q\ Z) -^

H^{Q\ R) = R

induced by /, we have /*[7r] = L • e{7T), where L = lcm[ai,..., ap]. Thus [ic] has infinite order in H^(Q; Z) if and only if e{7t) ^ 0 . We now characterize the cocompact orbifold groups modelled on P. THEOREM 6.3.5. An abstract group n can be embedded into Isomo(P) as a cocompact discrete subgroup if and only if n is a central extension of Z by a discrete cocompact orientation-preserving hyperbolic group Q (so that l-^Tj^jt-^Q-^lis exact) and [Ti:]e H^(Q;Z) has infinite order Further, if this is the case, the subgroup Z is the center of 71, and in any discrete embedding, the image oflj is TT Pi R, where R C R x^ P = Isomo(P).

PROOF. Suppose TT is a cocompact discrete subgroup of Isomo(P). The subgroup R of Isomo(P) is the radical (maximal connected normal solvable subgroup) of Isomo(P) and the quotient Isomo(IHI) = PSL(2, R) has no compact factor. A theorem of Wang (see [51, 8.27]) says that the image 2 of TT in Isomo(IHI) is a lattice so that 2 is a discrete cocompact orientation-preserving hyperbolic group. It remains to show that TT Pi R is non-trivial. Suppose not. Then TT is isomorphic to Q and hence, it has R-cohomological dimension 2. However, since n is cocompact in Isomo(P), its R-cohomological dimension is 3. This contradiction shows that TT HR = Z. Thus TT is of the form l - > Z ^ - 7 r - > 2 ^ ' 1- Clearly, Z is the center of TT since Q is centerless. We shall now sketch why [TT] must be of infinite order in H^(Q;Z). By Selberg's lemma, there exists a normal subgroup Qo of Q which is torsion free of finite index. Let 1 -^ Z ^- TTo ^^ 2o ^^ 1 be the pullback of the above exact sequence via 2o ^^ 2- But if

K.B. Lee and F. Raymond

692

[TT] has finite order, one can take Qo so that [TTQ] has order 0 so that TTQ = Z x (go- We claim that this group does not embed discretely into Isomo(P). Choose a standard presentation for (2o: a\,b\,...,ag,bg\

]~[[a/,Z7/] = 1

Since Qo C PSL(2, R), we may think of the a/, bi as elements of PSL(2, R). In Isomo(P), these elements lift to {(a-, ta^)}, {{b[, //,.)}, were a'-, b'. G P\ ta^Jbi ^ ^- These are unique up to the center of P. Since H L i [(^/^ ^a,), (b-, tbi)] = it^^~^, 0) by [55], it is non-zero. Since :7ro = Z x Qo C Isomo(P), this relation 2g — 2 would have to be 0. This gives a contradiction and so [n] must have infinite order. Conversely, suppose 1 ^ - Z — ^ j r ^ - Q - ^ l i s exact, where Q is a cocompact discrete subgroup of IsomoCH) = PSL(2, R); and [n] e H^iQ; Z) has infinite order. Then with the natural inclusion /: Z ^^-> R and the induced homomorphism i^\H^{Q\Ij) -^ H^{Q\ R), /*[7r] isjhe pushout [RTT] G H^{Q\ R ) (see Section 3.3.1), and is non-zero. By Lemma 6.3.3, [Q\ e H^{Q\ R) is also non-zero. Therefore, there exists ^ G R so that (e o /)^[7r] = ^*[R7r] = [2]. This implies that there exists a homomorphism of n into Q with the diagram 1 —^Z

^TT

Y

1^

Y

R —^ \s

>f

^ Q —^ 1

RTT

t

^

Y

G^

1

\ =

Y

1 - ^ R — ^ Q — ^ Q —^ 1 commutative and with injective vertical maps. Since Z and Q acts properly discontinuously with compact quotient on R and H respectively, n is cocompact and discrete. This completes the proof. D 6.3.6. Let p:Q-> PSL(2, R) be a discrete cocompact subgroup. For an extension l ^ Z - > 7 r - > 2 ^ 1» there exists an injective homomorphism 0 :7t -> Isom(P) so that the diagram COROLLARY

1

Z

- ^ 7T

Q

-^ 1

(6.3) -^ Isomo(P) — ^ PSL(2,R) — ^ 1 commutes if and only //"[TT] G / / ((g; Z) has infinite order.

Seifert manifolds

693

COROLLARY 6.3.7. (Structure) Let M be a closed orbifold modelled on (Isomo(P), P)geometry. Then M is an orientable closed Seifert orbifold over a hyperbolic base with

^(M)/0. (Realization) Let M be a compact orientable Seifert orbifold over a hyperbolic orbifold. Then M admits an (Isomo(P), P)-geometry if and only ife(M) ^ 0. 6.3.8. Consider the commuting diagram (6.3). With fixed e and p, how many ^'s are there to make the diagram conmiutative? Such maps are classified by //^ ( 2 ; M), see Theorem 3.7.2. Since the action of 2 on E is trivial,

where 2g is the first Betti number of the group Q. Compare this with //^((2;M(M,R)) = 0

(i^ 1)

so that, for any extension I ^ - Z ^ - T T ^ - Q ^^ l , a homomorphism n -^ TOP]i^(R x H) exists and is unique, up to conjugation by elements of M(IH, R).

6.4. Lorentz structures and PSL(2, W)-geometry The spaces admitting PSL2R-geometry have another interesting geometric structure. Here is a more explicit description of our problem. Let us denote PSL(2, R) by Poo. The space Poo is the universal covering group of P\ = PSL(2, R). Topologically Poo is homeomorphic to R X M. Pi=PSL(2,R), Poo=PSL(2,R). Consider the indefinite metric of signature + H

on R"^. The unit sphere of this space is

5^'2 ^ {(^^ 3;) I ^^ ^ ^ 1^2^ l^|2 _ |^|2 ^ 1 j ^ Q^2, 2 ) / 0 ( l , 2). The linear map of R"^ defined by the matrix 1 0 1 0 0 1 0 1 0 - 1 0 1 1 0 -1 0 transforms 5^'^ to P2 = {(x,y,z,u)eR^\xu-yz

= l} = SL2R.

694

K.B. Lee and F. Raymond

Thus P2 has a complete Lorentz metric of signature +, —, — and constant sectional curvature = 1. A space is called a Lorentz orbifold if it is the quotient of Poo by a discrete group of Lorentz isometries acting properly discontinuously. One can show that such a group contains normal subgroups of finite index which act freely. A Lorentz orbifold for which the discrete group acts freely is called a Lorentz space-form. The Lorentz structure on a space-form is non-singular and it has Poo as its (metric) universal covering. Then 1 —> Z -> Poo ^^ Pi ^^ 1 is actually a central extension with Z being the entire center of Poo- It turns out that the identity component of Isom(Poo) is (Poo x Poo)/^ where Z is the diagonal central subgroup corresponding to the center of each of the Poo-factors. The action Poo x Poo, as isometries, on Poo is given by (a, P)'X =a 'X - P- 1 Moreover, Isomo(Poo) = ^00 xz Poo has index 4 in Isom(Poo). These Lorentz space-forms are analogous to the complete spherical space-forms in the Riemannian case. Let us describe some obvious ones which turn out to be homogeneous in the sense that Isom(M) acts transitively on M. Take n C Poo x ^ C Poo x^ Poo as a discrete subgroup. Then surely the centralizer of n in Isom(Poo), C'isom(Poo)(^)' contains e x Poo in (Poo x^ Poo)- In fact, it is exactly e x Poo (unless TT ^ Z and sits in the center of Poo X ^ in Isom(Poo))- Such groups n are classified in [55] and are certain Seifert manifolds over a hyperbolic base, for there is an obvious S^ action on TT \ Poo induced from e X Poo C Isomo(Poo)- A surprising fact is that all homogeneous Lorentz orbifolds are actually homogeneous Lorentz space forms and coincide with those just described above, [30, §10]. If TT C Isom(Poo) so that M = n \ Poo is compact then it is shown in [30, §7] that M is homeomorphic to an orientable Seifert orbifold over a hyperbolic base. However, the connections between the Seifert structure and the Lorentz structure is unclear. This is due to the fact that Isom(Poo) does not act properly on Poo- By selecting a maximal subgroup of Isom(Poo) which acts properly on Poo, these two disparate structures can be related. The subgroup J (Poo) = (Poo xz M) x Z2 C Isom(Poo), where Poo x^ M = (Poo x R)/Z, where Z is the central diagonal subgroup of Poo x M, and Z2 reverses the orientation of time (= R) and space (= HI = Poo/^) at the same time is the same Lie group as the Isom(Poo) in the Riemannian case. A Lorentz orbifold (resp. spaceform) M = n \ Poo, where n C J (Poo) C Isom(Poo) is called standard. Therefore the standard Lorentz orbifolds and space-forms coincide with the orbifolds and space-forms of the PSL(2, M)-geometry. A Lorentz space-form is homogeneous if the full group of Lorentz isometries acts transitively. It is known that homogeneous Lorentz space-forms M admit nonstandard complete Lorentz structures if H\M; R ) ^ 0. See [18]. The Seifert fibering, in the theorem below, on M descends from the R-action by the second R-factor in J (Poo) on Poo- The following are due to Kulkami and Raymond and are stated here, for simpUcity, in the closed cases.

Seifert manifolds

695

THEOREM 6.4.1 ([30, (8.5)]). (Structure) A compact standard Lorentz space-form (resp. Lorentz orbifold) is an orientable Seifert manifold M {resp. Seifert orbifold) over a hyperbolic base B with e(M) ^ 0. (Realization). Let M be a compact orientable Seifert manifold (resp. orientable Seifert orbifold) over a hyperbolic base with e(M) ^ 0. Then, M admits a structure of a standard Lorentz space-form {resp. Lorentz orbifold).

We remark that the orbifold part breaks into two separate cases. If all fibers are ^ 5 ^ then all closed orientable Seifert manifolds with e{M) ^ 0 appear as Lorentz orbifolds (and conversely). This is similar to having the topological sphere appear as a 2-dimensional hyperbolic orbifold. If some fibers are arcs then the Lorentz orbifolds are homeomorphic to connected sums of lens spaces (including S^ and 5^ x 5'). We should also mention that the statements in Theorem 6.4.1 are for the full J {Poo) and not the connected component of the identity, and correspond to the cocompact discrete subgroups of Isom(P) in the Riemannian case instead of Isomo(P) as described in the preceding section. See [30, §8,9] for details and treatment of cases other than the compact ones.

6.5. Deformation spaces for PSL(2, R)-geometry 6.5.1. Let 7l{7t; U) be the space of all injective homomorphisms 0:n ^^U such that 0{7r) is cocompact acting properly on Poo and, is discrete in U. We topologize lZ{7i; hi) as a subset of W. In general, if ZY is a Lie group, then lZ{7t\ U) will be a real analytic space. DEFINITION

The space 7l{n\ hi) is called the space of discrete representations ofn into hi or the Weil space of{7T\ hi). When there is no confusion likely, we denote 1Z{7r;hi) simply by 1Z{7T). Recall that /x denotes conjugation. The inner-automorphisms group lnn{hi) acts on lZ{7t) from the left by /X(M)

• 0 = /X(M) OO

for u e hi and 0 e 7l{n). Denote the orbit space of this action by T{7t) =

lxm{U)\n{7t).

It is called the Teichmiiller space of n (or of M). Aut(7r) acts on lZ{'n) from the right by O'f

=

0of

for 0 G 7l{'n:) and / G Aut(7r). Denote the orbit space of this action by 5(7r)=7^(7^)/Aut(7r). S{7T) is the space of discrete subgroups of hi each isomorphic ton, or the Chabauty space.

696

K.B. Lee and E Raymond

Since the two actions of lnn(U) and Aut(7r) commute with each other, Aut(7r) acts on T(7r), and lnn(U) acts on S(n). Aut(7r) has an obvious kernel Inn(7r). Consequently, we get an action of Out(7r) on T(n). We denote the orbit space by M(7T)=T(7t)/Oui(7t). It is called the moduli (or Riemann) space of TT . It is also obtained as the orbit space A^(7r)=Inn(^/)\5(7r). Summarizing in the form of a commutative diagram of orbit mappings, we have _

lnn(U)\

_

(Inn(Z^), 7^(7^), Aut(7r)) — ^ (TCTT), Out(7r)) /Aut(7r)

/Out(7r)

(Inn(Z^),5(7r))

^""^^^^ ^

(6.4)

M(ji)

We shall now describe the deformation spaces for closed M which have a geometric structure modelled on (W, P) = (Isomo(/^), ^) (or equivalently the standard Lorentz structures). We shall see that these deformation spaces all have Seifert fiberings over well-studied deformation spaces of discrete cocompact orientation-preserving hyperbolic groups. Let TT be a cocompact discrete subgroup ofU = Isomo(Poo)- We have the central extension l - > Z ^ 7 r - > Q^

1,

[n]eH^(Q;Z) with 2 c P, and having infinite order mH^(Q;Z). Since our group U embeds into T O P M ( R X H ) we have topological rigidity in the strong sense that if 0\ and O2 are two embeddings of cocompact n into TOP]^(R x H), then they are conjugate in T O P K ( M X H ) . In almost all cases they will not be conjugate in U. The elements of M(7T) then represent the different ^/-structures on M, and we may expect large deformation spaces. In order to understand 1Z(7t), we need to study Aut(7r). We would like to describe Aut(7r) in terms of Aui(Q). Since Z is characteristic, any automorphism of n induces an automorphism of Q. DEFINITION 6.5.2. Let Aut(2(TT)) betheimageof Aut(7r) -^ Aut(2). Thatis, Aui{Q(7t)) = {0 e Aut((2): 30 e Aui(n) inducing (9}, the group of automorphisms of Q which can be lifted to an automorphism of TT.

Seifert manifolds

697

Because Z C M has the unique isomorphism extension property, one can form a pushout (see Section 3.3.1) to get RTT fitting the commuting diagram: 1

Z

1

1 LEMMA

Q

-^ 1

6.5.3. There is a commutative diagram with exact rows and injective vertical

maps: 1

1

Hom((2,Z)

Aut(7r)

1

HomCe,]

-^ Aut(R7r)

- Aut(e(;r))

1

I ^ Aui(Q)

(6.5) -

-^ 1

The crucial fact for the proof is H^(Q;R) = R. Since [7T]eH^(Q; Z) has infinite order, [RTT] G i / ^ ( 2 ; R) is non-zero. Since R is characteristic in RTT, any / e Aut(R7r) induces an automorphism / e Aui(Q). Suppose / = id. Let / : R ^- R be the restriction of / . Then / J R T T ] = /*[R7r] = [RTT] since / = id. Since [RTT] is non-zero, / = id. Therefore, / is of the form PROOF.

f(a) = k(a) ' a for some map A: TT ^- R. One easily sees that X factors through Q and it satisfies the cocycle condition

for all a, /3 G <2 so that X e Z^(Q,W). However, since R is central in RTT, R is a trivial Q-module so that Z\Q, R ) = Hom(Q, R). Conversely, any such a A G Hom((2, R) yields an automorphism / G Aut(R7r). Moreover, Hom(2, R) fl Aut(7r) = Hom(Q, Z). Let g G Aui{Q). We would like to find g G Aut(R7r) which induces g on Q. The automorphism g induces an automorphism g"": H^(Q;R) ^ H^(Q;R). Since H^(Q;R) = R and [RTT] ^ 0, there is a real number e for which g*[R7r] = ^[R:7r]. This number e is non-zero and can be viewed as an automorphism of RTT such that 1 --^

R —^ £

^ 1 -- ^ i I—^R

RTT

•

—^ Q —^ 8

n ^ i 2 —^

is commutative. Thus we have shown that Aut(R7r) -^ Aut((2) is surjective.

D

The deformation spaces of n will be studied via those of (2- To this end, it is necessary to define the following:

698

K.B. Lee and F. Raymond

DEFINITION 6.5.4. Let 7^(2; 2/^) be the space of all injective homomorphisms 0(Q) such that 0(Q) is cocompact and discrete in U. There is a left action of lnn(U) on 1Z(Q; U), and also a right action of Aut(2(7r)) on 7^(|2;Z7). We then define

r(Q;U)

=

lnn(l()\n(Q'M,

S{Q(7t); U) = n(Q; M{Q(7ty, U) = T{Q-

U)/Aut{Q(7t)), U)/0ut{Q(7t)).

These are the Teichmuller space, the restricted Chabauty space, and the restricted moduli space of 2 , respectively. It is known that Aut((2(7r)) is a subgroup of Aut((2) of finite index, see [29, Appendix]. This implies that S{Q{7T)) = 1Z(Q)/Aut(Q(7t)) is a finite regular covering of S(Q). Also M(Q(7T)) = r(Q)/0ut(Q(7i)), where Out(e(7r)) = Aut(e(7r))/Inn(e). 6.5.5. The action of Aui(7t) on 1Z{7T\U) extends to an action o/Aut(R7r) on n{Tt\U), The subgroup Z\Q\ R ) = H\Q\W) = Hom((2, R) acts on IKji'M) freely and properly. Moreover, 7^(7^; Z^)/Hom(2, R) = 1Z{Q\ U). LEMMA

PROOF. Clearly, an element 0 e 1Z{IT\U) determines a homomorphism Q-.RTT -^ U uniquely. The action of Aut(R7r) on 7^(7r; U) is defined as follows: for Q € 1Z(n;U) and / G Aut(R7r),

O'f

=

0of\,,

The image of n under ^ o / is a discrete cocompact subgroup of U so that 0 o f\jj^ e n(7t;U), _ Suppose 0,0' e 71{7T\U) induce the same representation 0 = 0' G 1Z{Q\U). Let 0,0' \^TC -^ hi ht the homomorphisms induced from 0,0' as described above. Since 0=0', the embeddings 0 and 0' are related by d'(a) = X(a)0(a) for some map X: RTT ^• RcU. Since 0 and 0' must be equal on R, the map k factors through Q, and hence X:Q—>R. The map X satisfies the cocycle condition so that X e Kom(Q,R). Conversely, let / G Aut(R7r) inducing the identity on Q. Then by reversing the order of arguments given above, one sees that 0 and Oof represent the same element of 1Z(Q;U). Clearly, unless / is the identity, 0 and Oof will be different, which shows that the action of Hom((2, R) on 7Z(7T;U) is free and proper since the orbit space is Hausdorff. D 6.5.6 ([29, Theorem 2.5]). Let n bea compactorbifoldgroup with (Isomo(P), P)-geometry. Let g be the genus of the base orbifold. Then,

THEOREM

n(7t) = n(Q)x

R^^ trivial principal R^^-bundle over Ti{Q),

Seifert manifolds

699

T(7r) = T(|2) X R^^ trivial principal ^^^-bundle over T(Q), S(7r) T^^-bundle over S(^Q{n)), Ai{n) Seifert fiber space over Ai(^Q(7r)) with typical fiber T^^. Furthermore, S(Q(7t)) is a finite sheeted covering of S{Q), and sheeted branched covering ofM.{Q).

M{Q(7T))

is a finite

PROOF. Let Z be the center of n. Then Q = n/Z is the base orbifold group. Hence in Lemma 6.5.3, Hom(2, R) = H\Q;R) = R^^, and

1^

R2^ ^

Aut(R7r) -^

Aut(Q) -^

1

is exact. By Lemma 6.5.5, the group R^^ acts on lZ(n) freely and properly so that the orbit map becomes a principal bundle >7^(7r)->7^(e). Since R^^ is contractible, its classifying space is a point and consequently lZ(n) splits as (n(n), R^^) = (7^(2) X R2^, R2^) equivariantly, where R^^ acts on the second factor as translations. Since R is the center of Isomo(P), we have /ji(lsomo(P)) = ii{P) and T{7T) = /x(lsomo(P)) \ 7^(7r) = /x(P) \ {n(Q) x R^^). Now /jiiP) acts on 1Z(Q) freely and properly with quotient T(Q) [44] which has the homotopy type of the set of two points. Therefore T(7r) is a product T(Q) x R^-^. Moreover, it is known that T(Q) is diffeomorphic to two copies of R^^~^+2^, where p is the number of non-free orbit types of the Q action on H. (This is the number of distinct conjugacy classes of maximal finite subgroups of Q.) For the space of subgroups S(n) = 7^(7^)/Aut(7^) = (7^((2) x R2^)/Aut(7r), note that Aut(7r) n R^-^ = Z^^ and the quotient Aut(7r)/Z2^, which we called Aui(Q(7t)), is a subgroup of Aut(Q). By first dividing out by Z^^, we get iS(7r) = (Tl(Q) x T^^)/Aui(Q(7t)). Since Aut(2(7r)) acts on 1Z(Q) freely, we have a genuine fibration

The action of Aut(7r) on R^^ xTl(Q) = Tl(n) is weakly R^^-equivariant, because Z^^ = Hom(2; Z) is normal in Aut(7r). In other words, Aut(7r) ^

TOP3^2,(R^^ x7^(e))

= M(7^(G),R2^) >^ (GL(2g,R) xTOP(7^(e)))

700

K.B. Lee and F. Raymond

so that 1

^j}s

^ Aut(7r)

Y

^ Aut((2(7r))

Y

^1

Y

1 - ^ M ( 7 ^ ( | 2 ) , M^^) — ^ T O P K 2 , ( R 2 ^ X 7^(e)) —^GL(2g, E) x T0P(7^(e)) —^ 1 is commutative. Thus, the structure group is a subgroup of the affine group of the torus T^^ o GL(2g, Z). Let 0 e 7^(7^). Then for a G TT, 6> • fi(a) = /x(6>(Qf)) o 0. Therefore, on T(7r), Inn(7r) = Inn(R7r) = Q acts trivially as does Inn(Q) = Q on T(Q), Consequently, we have properly discontinuous actions of Out(lR7r) and Out(Q(n)) on T(7r)

andT(e). The space of moduH M(n) = T(7r)/Out(7r) requires more care. Recall that 7Z(7r) = M?^ X T(Q) with R^^ action by translations on the first factor. Since 1 —> R^^ ^

Out(R7r) —> Oui(Q) -^

1

is exact, we also have the commutative diagram: (T(7r)=R2^ xT(e),0ut(R7r)) — ^ (T(e),Out(G)) /Out((2(7r))

/Out(7r)

X(7r) = T(7r)/Out(7r) —

^

(6.6)

M{Q(it))

The actions and maps arise from the embedding 1 — ^ j}^ Y

^ Out(7r)

^ Out((2(7r)) — ^ 1

Y

1 — ^ R2^ — ^ Out(R7r)

(6.7)

Y

^ Out((2)

^ 1

obtained from Lemma 6.5.3 by dividing out the ineffective Q. Note Inn(R7r) O R^^ = 1. Now as Out(7r) normalizes R^^ and 1?^ sits in R^<^ as a lattice, the mapping ^ is a Seifert fibering with typical fiber the torus R^^/Z^^. In general, the fibering will not be locally trivial. In fact, if F = (Out(g(7r)))^^j for some [^] G T((2), then the induced extension 1^

^2g ^

£ _ ^ 77 — ^ 1

acts affinely on R^^ x [^] sitting over \d\ The orbit over \d\ under Out((2(7r)) determines a 2-dimensional hyperbolic orbifold up to isometry. Over this hyperbolic orbifold is the set F \ R^^ = F \ T^^ of metric Seifert orbifolds in Miji) with base this hyperbolic orbifold. •

Seifert manifolds

701

6.6. Polynomial structures on virtually poly-IJ manifolds We have seen in Theorem 5.2.3 that any torsion-free virtually poly-Z group n is the fundamental group of a closed K{7T, l)-manifold. For example, see (4.1). On the other hand, Milnor [46] has shown that such a group n can be the fundamental group of a complete affinely flat (not necessarily compact) manifold. The question arises as to whether one can find a compact complete affinely flat manifold with such fundamental group. It is true when 7t is 3-step nilpotent [57]. However, counter-examples produced by Benoist [5] and Burde and Grunewald [10] show that certain compact nilmanifolds do not admit a complete affinely flat structure. Consequently, the question has a negative answer even in the nilpotent case. Note that one can look at such a complete affinely flat structure as a "polynomial structure of degree 1". In a situation where "polynomial structures of degree 1" fails to exist, the next best structure will be "polynomial structure of higher degree". The main reference for this section is [14]. A torsion-free filtration for a torsion-free finitely generated nilpotent group T is a central series of the form: r*: To = 1 c Ti c r2 c • • • c rc-\ CFCC r^+i = r for which Fi+i/Fi = Z^

for 1 ^ / ^ c and some k G NQ.

Moreover, each Fi can be chosen to be a characteristic subgroup of 7^. We use K to denote the Hirsh number (or rank) of F. Often, we will also use Ki = ki + /:/+i + • • • + /c^- It follows that/^ = /ri. NOTATION 6.6.1. (1) P(R^, R^) c M(R^, R^) denotes the vector space of polynomial mappings p: R^ -^R^. So p is given by k polynomials in K variables. (2) P(R^) C TOP(R^) will be used to indicate the set of all polynomial diffeomorphisms of R^, with an inverse which is also a polynomial mapping. This is a group where the multiplication is given by composition.

It is not hard to verify that P(R^,R^) is a Aut(Z^) x P(R^)-module and that the resulting semi-direct product group P(R^,R^) X (Aut(Z^) X P{R^)) c P(R^+^) by defining V(/7, g, h) e P(R^, R^) x (Aut(Z^) x P(R^)), V(jc, y) e R^+^:

^P^^^^\x,y) =

{gx-p{h(y)),h(y)),

Restrict M(R^, R^) to P(R^, R^) and TOP(R^) to P(R^), and we will speak of canonical type polynomial representations.

702

K.B. Lee and F. Raymond

From now on, if we speak of a polynomial representation p: F ^^ P(R^) which is of canonical type, we mean of canonical type with respect to some torsion-free central series. THEOREM 6.6.2. (Existence) Let F be a torsion-free, finitely generated nilpotent group of rank K. For any torsion-free central series F^, there exists a canonical type polynomial representation p-.F^- P(R^). (Uniqueness) If p\,p2 cire two such polynomial representations of F into P(R^) with respect to the same F^, there exists a polynomial map p e P(R^) such that p2 = p~^ o p\o p. Consequently, the manifolds p\ (F) \ R^ and P2(F) \ R^ are ''polynomially diffeomorphic ". PROOF.

The fundamental fact that we shall use is the cohomology vanishing //'•(r/ri,p(R^2^R^')) = o

for / > 0. See [14] for a proof. (Existence) We will proceed by induction on the nilpotency class c of F . If T is Abelian, then the existence is well known. Now, suppose that F is of class c > 1 and the existence is guaranteed for lower nilpotency classes. Using the induction hypotheses, the group F/F\ can be furnished with a canonical type polynomial representation

p-r/Fx^ We obtain an embedding /: A = Z^i -> P(R^2 R^O if we define /(z) :R^2 _^ j x\-^ z. We are looking for a map p making the following diagram commutative: 1

^ Fx Y

^ F

^ F/Fx

Y

^ 1

Y

1 — ^ P(R^2^R^i) — ^ P X (A X P) — ^ Aut(Z^i) X P(R^2) — ^ i

n K

)

where P x (A x P) denotes P(R^^R^i) x (Aut(Z^O x P(R^2)) and (p\F/Fx -^ Aut(Z^') = Aut(ri) denotes the morphism induced by the extension 1 ^- Ti -> T ->

r/Fx ^ 1. The existence of such a map p is now guaranteed by the surjectiveness of 5 in the long exact cohomology sequence

>0 = -^ H^{F/Fx ,1^')—^

H\FIFx,P{pJ^\^^'))--^H\F/Fx.P{^^\R^')ll}') H^{F/Fx, P(R^2^ R^')) = 0 - ^ • • •.

(Uniqueness) Again we proceed by induction on the nilpotency class c of T. For c = 1 the result is again well known. Indeed, two canonical type representations of a virtually Abelian group are even known to be affinely conjugate.

Seifert manifolds

703

So we suppose that F is of class c > 1 and that the theorem holds for smaller nilpotency classes. The representations pi, P2 induce two canonical type polynomial representations

By the induction hypothesis, there exists a polynomial map q : R^^ _^ ]^^2 such that P2=q~^

opioq.

Lift this ^ to a polynomial map q of R^^ as follows: Vx G R^», Vy G R^2. ^ : M^i ^ R^i : (j^, >;) \-> q(x, y) = {x, q(y)). When restricted to Fi, p\ and p2 are mapping elements onto pure translations of R^^. We know that there exist an affine mapping A : R^i -^ R^i for which

Extend A to an affine mapping of R^ by defining VxGR^^V};GR^2. A:R^i ^ R^i :(x,y)\-^

A(x,y) =

{A(x),y).

Let us denote xfr = A~^ oq~^ o p\ oq o A. Then we see that \l/ and P2 are two canonical type polynomial representations of F, which coincide with each other on F\ and which induce the same representation of F/F\. This means that V^ and p2 can be seen as the result of a Seifert construction with respect to the same data (cf. the commutative diagram above). Now we use the injectiveness of 8, which implies that i/^ and p2 are conjugate to each other by an element r e P(R^^, R^') (seen as an element of P(R^)!). So we may conclude that p2 = r~ o\j/or = r~ o A~ o q~ o p\ oq o A o r = p~ o p\ o p if we take p = q o A or.

D

References [1] A. Adem and J.F. Davis, Topics in transformation groups. Handbook of Geometric Topology, R.J. Daverman and R.B. Sher, eds, Elsevier, Amsterdam (2002), 1-54. [2] L. Auslander, Bieberbach's theorem on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. 71 (3) (1960), 579-590. [3] L. Auslander, On radicals of discrete subgroups of Lie groups, Amer. J. Math. (1963), 145-150. [4] L. Auslander and F.E.A. Johnson, On a conjecture of C.T.C. Wall, J. London Math. Soc. (2) 14 (1976), 331-332. [5] Y. Benoist, Une nilvariete non affine, C. R. Acad. Sci. Paris Ser. 1315 (1992), 983-986.

704

K.B. Lee and F. Raymond

[6] A. Borel et al.. Seminar on Transformation Groups, Ann. of Math. Stud., Vol. 46, Princeton University Press (1960). [7] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, Inc., Orlando, FL (1972). [8] W. Browder and W.C. Hsiang, G-actions and the fundamental group. Invent. Math. 65 (1982), 411^24. [9] K.S. Brown, Cohomology of Groups, Grad. Texts in Math., Vol. 87, Springer, New York (1982). [10] D. Burde and F. Grunewald, Modules for certain Lie algebras of maximal class, J. Pure Appl. Algebra 99 (1995), 239-254. [11] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press (1956). [12] P.E. Conner and F. Raymond, Actions of compact Lie groups on aspherical manifolds. Topology of Manifolds, J.C. Cantrell and C.H. Edwards, eds, Markham, Chicago (1969), 227-264. [13] P.E. Conner and F. Raymond, Holomorphic Seifert fiberings, Proc. 2nd Conference on Transformations groups. Part II, Lecture Notes in Math., Vol. 299 (1971), 124-204. [14] K. Dekimpe, P. Igodt and K.B. Lee, Polynomial structures for nilpotent groups. Trans. Amer. Math. Soc. (1996), 77-97. [15] H. Donnelly and R. Schultz, Compact group actions and maps into aspherical manifolds. Topology 21 (1982), 443^55. [16] G. Dula and D. Gottlieb, Splitting off H-spaces and Conner-Raymond splitting theorem, J. Fac. Sci. Univ. Tokyo Sect. lA Math. 37 (2) (1990), 321-334. [17] F.T. Farrell and W.C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds M'' (n ^ 3,4), Amer. J. Math. 105 (1983), 641-672. [18] W. Goldman, Nonstandard Lorentz space forms, J. Differential Geom. 21 (1985), 301-308. [19] D. Gottlieb, Splitting off tori and the evaluation subgroup of the fundamental group, Israel J. Math. (1989), 216-222. [20] D. Gottlieb, K.B. Lee and M. Ozaydm, Compact group actions and maps into kin, l)-spaces. Trans. Amer. Math. Soc. (1985), 419^29. [21] H. Holmann, Seifertsche Faseraume, Math. Ann. 157 (1964), 138-166. [22] P. Igodt, Generalizing a realization result ofB. Zimmermann and K.B. Lee to infranilmanifolds, Manuscripta Math. (1984), 19-30. [23] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), 235-265. [24] F.W. Kamber and Ph. Tondeur, Flat manifolds with parallel torsion, J. Differential Geom. 2 (1968), 385389. [25] E. Kang and K.B. Lee, Injective operations of homogeneous spaces, Michigan Math. J. (1997), to appear. [26] Y. Kamishima, K.B. Lee and F. Raymond, The Seifert construction and its applications to infranilmanifolds. Quart. J. Math. Oxford 34 (1983), 433^52. [27] H. Kim, Complete left-invariant affine structures on nilpotent Lie groups, J. Differential Geom. (1986), 373-394. [28] H.T. Ku and M.C. Ku, Group actions on aspherical A^ {N)-manifolds. Trans. Amer. Math. Soc. 278 (1983), 841-859. [29] R. Kulkami, K.B. Lee and F. Raymond, Deformation spaces for Seifert manifolds. Geometry and Topology, University of Maryland 1983-1984, Vol. 1167, J. Alexander and J. Harer, eds (1986), 180-216. [30] R. Kulkami and F. Raymond, 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. (1985), 231-268. [31] J.B. Lee, K.B. Lee and F. Raymond, Seifert manifolds with fiber spherical space forms. Trans. Amer. Math. Soc. 348 (1996), 3763-3798. [32] K.B. Lee, Geometric realization ofnQsim), Proc. Amer. Math. Soc. 86 (2) (1982), 353-357. [33] K.B. Lee, Geometric realization ofnosim) II, Proc. Amer. Math. Soc. 87 (1982), 175-178. [34] K.B. Lee, Aspherical manifolds with virtually 3-step nilpotent fundamental group, Amer. J. Math. 105 (1983), 1435-1453. [35] K.B. Lee, Infra-solvmanifolds of type (R), Quart. J. Math. Oxford (2) 43 (1995), 185-195. [36] K.B. Lee and F. Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geom. 16 (1981), 255-269. [37] K.B. Lee and F. Raymond, Topological, affine and isometric actions on flat Riemannian manifolds II, Topology Appl. 13 (1982), 295-310.

Seifert manifolds

705

[38] K.B. Lee and F. Raymond, Geometric realization of group extensions by the Seifert construction, Contemp. Math. 33 (1984), 353^11. [39] K.B. Lee and F. Raymond, Rigidity of almost crystallographic groups, Contemp. Math. 44 (1985), 73-78. [40] K.B. Lee and F.A. Raymond, Manifolds on which only tori can act. Trans. Amer. Math. Soc. (1987), 487499. [41] K.B. Lee and F. Raymond, Seifert manifolds modelled on principal bundles, Transf. Groups (OsaJca, 1987), Lectures Notes in Math., Vol. 1375 (1989), 207-215. [42] K.B. Lee and E Raymond, Seifert manifolds with r\G/K-fiber, Michigan Math. J 43 (1996), 437-464. [43] S. MacLane, Homology, Grundlehren Math. Wiss., Vol. 114, Springer, Berlin (1975). [44] A.M. Macbeath and D. Singleman, Spaces of subgroups and Teichmiiller spaces, Proc. London Math. Soc. (3) (1975), 211-256. [45] A. Mal'cev, On a class of homogeneous spaces. Trans. Amer. Math. Soc. 1 (9) (1962), 276-307. [46] J. Milnor, On fundamental groups of complete affinely flat manifolds. Adv. Math. 25 (1977), 178-187. [47] A. Nicas and C. Starlc, K-theory and surgery of codimension-two torus actions on aspherical manifolds, J. London Math. Soc. 31 (2) (1985), 173-183. [48] R Odik, Seifert Manifolds, Lecture Notes in Math., Vol. 291, Springer (1972). [49] R. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math (1961), 295-323. [50] C.Y. Park, Homotopy groups of automorphisms of some injective Seifert fiber spaces. Thesis, University of Michigan (1989). [51] M.S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb., Vol. 68, Springer (1972). [52] E Raymond, The Nielsen theorem for Seifert fiber spaces over locally symmetric spaces, J. Korean Math. Soc. 16 (1979), 87-93. [53] E Raymond and L. Scott, The failure of Nielsen's theorem in higher dimensions. Arch. Math. 29 (1977), 643-654. [54] E Raymond and D. Wigner, Construction of aspherical manifolds, Geom. Appl. Homotopy Theory 657 (1977), 408^22. [55] F.A. Raymond and A. Vasquez, 3-Manifolds whose universal coverings are Lie groups. Topology Appl. 12 (1981), 161-179. [56] H. Rees, Special manifolds and Novikov's conjecture. Topology 22 (1983), 365-378. [57] J. Scheuneman, Affine structures on three-step nilpotent Lie algebras, Proc. Amer. Math. Soc. 46 (3) (1974), 451-454. [58] P Scott, The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), 401-487. [59] R. Schoen and S.T. Yau, Compact group actions and the topology of manifolds with non-positive curvature. Topology 18 (1979), 361-380. [60] H. Seifert, Topologie drei-dimensionaler gefaserter Rdume, Acta. Math. 60 (1933), 147-238. [61] E Waldhausen, Gruppen mit Zentrum und drei-dimensionaler Mannigfaltigkeiten, Topology 6 (1967), 505517. [62] R. Washiyama and T. Watabe, On the degree of symmetry of a certain manifold, J. Math. Soc. Japan 35 (1983), 53-58. [63] H. Zieschang, On toric fiberings over surfaces. Math. Notes (1969), 341-345. [64] H. Zieschang and B. Zimmermann, Endliche Gruppen von Abbildungsklassen gefaserter 3-Mannigfaltigkeiten, Math. Ann. (1979), 41-62. [65] B. Zimmermann, Uber Gruppen von Homoomorphismen Seifertscher Faserraume und flacher Mannigfaltigkeiten, Manuscripta Math. 30 (1980), 361-373.

This Page Intentionally Left Blank

CHAPTER 14

Quantum Invariants of 3-Manifolds W.B. Raymond Lickerish Department of Pure Mathematics and Mathematical Statistics, C.M.S., Wilberforce Road, Cambridge, CBS OWB, England, UK E-mail: wbrl@dpmms. cam. ac. uk

Contents 1. Introduction 2. The Kirby calculus 3. Skein theory based on the Kauffman bracket 4. Invariants of 3-manifolds from skein theory 5. Some applications 6. Fusion 7. The state sum invariants of Turaev and Viro 8. Invariants from HOMFLY-based skeins 9. Conclusion References

709 710 711 717 720 723 727 729 733 733

HANDBOOK OF GEOMETRIC TOPOLOGY Edited by RJ. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

707

This Page Intentionally Left Blank

Quantum invariants

709

1. Introduction The theory of quantum invariants of 3-manifolds is in a strange unsatisfactory state. On the one hand that makes it exciting and yet it is also somewhat bewildering. Descriptions of the invariants sometimes require considerable prior knowledge in some speciality. The invariants obtained by different routes tend to differ, by various normalisations and constants, and that makes translation arduous. Very many people are working in and around this subject. For many of them 3-manifold theory itself is not really a major interest, for these invariants can be interpreted from view-points in Quantum Field Theory, Statistical Mechanics, Category Theory, Differential Geometry, Operator Algebras and probably several more. Within the ambiance of these invariants the new subjects of Quantum Groups [10,43] and Topological Quantum Field Theory [1] have been bom; a recent survey is in [42]. This article tries to introduce one version of some of these quantum invariants to mathematicians who already have an interest in low-dimensional topology. It makes no attempt to unify vast areas of human knowledge, to search for ever deeper truth nor to develop the most sophisticated notations. The main emphasis will be on proving the existence of some non-trivial quantum invariants using the Unear skein theory associated to the Kauffman bracket; this method was developed progressively in [23-25,27]. That method has the advantages of being fairly quick, self-contained and visualisable. A few calculations of the invariants will be made together with intimations of how other calculations might be made. It should be admitted, at the start, that, from the perspective of the established theory of 3-manifolds, these invariants have been disappointing. Applications are few and far between. Certainly the invariants do distinguish apart various pairs of 3-manifolds but such pairs were already known to be distinct. Sadly, the invariants fail to classify lens spaces. The invariants that will be described here are in some sense generalisations of the original Jones polynomial. That polynomial invariant has had great success in giving information about ways of drawing diagrams of knots. To find some similar relevance of the quantum invariants to any facet of standard 3-manifold theory has become a provocative challenge to low-dimensional topologists. It is instructive to start at the beginning. According to legend, whilst dining with Sir Michael Atiyah in Annie's Cafe in Swansea, Ed Witten produced the following definition. Zk(M)=

f e?^/Mtr(dAAA + fAAAAA)^^^ JAeA lAeA

This was to be the definition of an invariant Zk{M) of a closed orientable 3-manifold M, the invariant depending upon a 'level' k and a Lie group G with Lie algebra g; A denotes the moduli space of connections on a G-bundle over M. The expression /^tr(dA A A + | A A A A A) is a multiple of the famous Chern-Simons action on A, where the connection A is now regarded as a g-valued 1-form. If the main integrand were multiplied by a monodromy term

n.„(.xp(/A))

710

W.B.R, Lickorish

this would become an invariant of a framed link L, with components Li, L 2 , . . . , L„, contained in M with each L/ equipped with an irreducible representation pi of g. The difficulty however is that the main integral over A has never been satisfactorily defined in spite of much effort. Nevertheless formal manipulations of this integral have led to results later rigorously estabhshed. In particular, when M is the 3-sphere S^, G = SU(2) and L is a 0-framed link equipped with the 2-dimensional representation on every component, then the invariant becomes the Jones polynomial VL (t) evaluated at e x p ( | ^ ) . Furthermore, the invariant should interact well with the idea of surgery. The first proofs that invariants exist in accord with these ideas were given given by Reshetikhin and Turaev [39] for SU(2) and Turaev and Wenzl [45] for other Lie groups using representations of universal enveloping algebras with certain roots of unity for their parameter q. 2. The Kirby calculus All proofs of the existence of quantum invariants seem to start with a 3-manifold given in terms of surgery on a framed link in 5^. A framed link L in S^ is a link together with a never-zero cross-section of its normal bundle. Thus L can be thought of as a link of bands (long thin annuli), two such links being equivalent if the bands of one can be moved to those of the other (by ambient isotopy). If one orients the two boundary components of a band, in the same direction around the band, and considers their linking number, that gives an integer framing number that specifies the framing of the component considered. The framing also specifies up to isotopy a simple closed curve (a boundary component of the band) on the boundary of a tubular neighbourhood N(Li) of each component L/. That is called the framing curve of L/. The oriented 3-manifold M obtained from S^ by surgery on the framed link L is given by DEFINITION.

M = 53 - A^(L) U (solid tori), where one solid torus is glued by its boundary to each dN(Li) in such a way that a curve bounding a disc in the solid torus is identified with the framing curve. A classical result about 3-manifolds is, then, the following [21,46]. 2.1. Any closed connected oriented 3-manifold can be obtained from S^ by surgery on some framed link. THEOREM

It is easy to give examples [22] in which surgeries on different framed knots gives the same 3-manifold, and even easier to do this for links. It is the next result of Kirby [12] that explains necessary and sufficient conditions for this to happen. 2.2. Two framed links specify by surgery the same 3-manifold, up to orientation preserving homeomorphism, if they differ by (ambient) isotopy in S^ and a sequence of moves of the following two types. THEOREM

Quantum invariants

111

Fig. 1.

K(l) Add or subtract an unknot with framing ±1 that is contained in a ball otherwise disjoint from the links. K(2) Suppose two components of a framed link and their framing annuli lie, as in one of the diagrams of Figure 1, in a doubly punctured disc that is contained in S^ (in maybe a very knotted, twisted, linked-up way). Replace those two framed components by the two in the other diagram of Figure 1 in the same doubly punctured disc. This theorem is often viewed in terms of adding and sHding 2-handles attached to a 4-ball, the boundary of the whole ensemble being the given 3-manifold. The proof of the theorem in [12] is an application of Cerf Theory to such 4-manifolds. These two results mean that, to define a 3-manifold invariant, it is just necessary to associate some mathematical entity to a framed link and then prove it invariant under moves of types K(l) and K(2). As an easy example, orient a framed link L and consider the matrix A of linking numbers, Aij = lk(L/, Lj), with Atj being the framing number of the component L/. It is easy to see that the Abelian group presented by this matrix of integers is invariant under the K-moves. However this group is easily shown to be the first homology group of the 3-manifold obtained by surgery on L. The idea of what follows is to use some polynomial invariant of the framed link L. It is not quite that easy however. What does work is a linear combination of the Jones polynomials, evaluated at certain complex roots of unity, of links formed by adding various parallels to the components of L. That, together with an adjustment depending only on the signature of the above linking matrix A will give a 3-manifold invariant. The complications, needed in defining the coefficients in the linear combination, are eased by the systematic approach of linear skein theory which will now be described. 3. Skein theory based on the Kauffman bracket Let M be an oriented 3-manifold with a finite set (maybe empty) of framed points in its boundary. The framing of such a point x is a specified element of the tangent space ofdM at X. Let A be a fixed complex number; later it will be a root of unity. DEFINITION. The linear skein space SM is the vector space over C generated by all links of framed simple closed curves and framed proper arcs in M, that meet 9M in precisely the given framed points, quotiented by the three relations of Figure 2.

Here (i) means that the union of L with a 0-framed unknot in a ball disjoint from L is equal to(—A~^ — A^)L. The equality of (ii) refers to three framed links that are identical

712

W.B.R. Lickerish

(i)

O

m (ill)

^

/

/

'^

=

(-A-IA')L

_

Isotopy of framed links. Fig. 2.

Type 11

Type III Fig. 3.

except in a ball in which they are as shown, all framing directions pointing towards the eye of the reader. As an exercise prove that, if a framed link L is changed to L' by adding one complete right-hand twist to the framing of one component, then L' = —A^L in SM. Elements of SM are sometimes called skeins in M. First consider SS^. All links can be projected to become link diagrams in S^ with the framings projecting to be the normal to 5'^. In this way a link diagram represents SL framed link. Ambient isotopy of framed links in S^ translates to equivalence of diagrams by Reidemeister moves of types II and III, see Figure 3 (and homeomorphisms of S^). A few moments thought will show that quotienting by (i) and (ii) implies the quotienting by these two types of Reidemeister move. Thus SS^ is isomorphic to the space spanned by link diagrams quotiented by (i) and (ii). Then very simple induction arguments concerning the number of crossings in a diagram show that SS^ is a 1-dimensional vector space; the empty link will be taken to be a natural base. To find the coordinate of a framed link, represent it by a diagram with n crossings, use (ii) to express it as a sum of 2^ diagrams with no crossing, then in each diagram remove all the resulting curves with no crossing by using (i). This is, of course, the essence of the Kauffman bracket approach to the Jones polynomial. If the 0-framed unknot were taken as base, the coordinate of a 0-framed knot would be that knot's Jones polynomial int = A~^ (where A is here being regarded as an indeterminate, not a complex number). The same method applies to the product of any surface F with an interval. The skein space has a base of framed links represented by all link diagrams in F that have no crossing and no null-homotopic curve. Thus a solid torus, parametrised as 5^ x D^, projects to the annulus S^ x I where / is a diameter of D^. Then S(S^ x D^) has, as a base, framed hnks represented by diagrams of the form S^ x X c S^ x I where X is a finite set. Now take a pair of disjoint orientation-preserving embeddings of D^ into D^ and take the product

Quantum invariants

713

= W'lM Fig. 4.

/

(n)

^

Fig. 5.

with the identity on S^ to obtain an embedding {s'xD^)u{s'xD^)-^{S'xD^), where 'u' denotes disjoint union. This induces, by embedding pairs of framed links, a bilinear map S{S^ X D^) X S{S^ X D^) -^ S{S^ X D^), Under this bilinear map S(S^ x D^) becomes a commutative algebra with just one generator, which will be called a, the framed link of one component represented by (S^ X one point) cS^ x I. Thus S(S^ x D^) is the polynomial algebra C[a]. A more subtle example of this skein theory is that of the 3-ball with In specified framed boundary points. That will be considered in terms of projections to link diagrams of arcs and closed curves in a rectangle, with n points on its left side and n on its right side and all framing directions pointing towards the viewer. As before that space has a base consisting of all diagrams of n arcs, with no crossing, joining up the In boundary points. The dimension of the space is the Catalan number ^ ^ (^"). However, juxtaposition of rectangles, identifying the right side of one with the left side of the next, at once induces a product on the space that makes it a (non-commutative) algebra. This is TLn, the n-\h Temperley-Lieb algebra. As an algebra it is generated by just the n elements, {l,e\,e2,.. .,en-\} that are shown in Figure 4. That diagram introduces the convention, to be used throughout, that an integer n beside an arc means that n parallel copies of the arc are present. Of course, 1 is a multiplicative identity of the algebra. Now will be defined, by induction on n, the all important Jones-Wenzl (see [47]) idempotent /^"^ G TLn. It is characterised in Lemma 3.1. In diagrams that ensue, a small white square will represent this /^"^; the number of arcs entering that square determines to which n it refers. Suppose (the linear sum of diagrams that represents) /^"^ is placed in S^ and its 2n specified boundary points are joined up by n parallel standard arcs, as in Figure 5. The resulting element of S(S^) = C so represented will be denoted An.

714

W.B.R. Lickorish

p+7

i+i

Fig. 6.

' n-V

»«.!

CP^i hCK

«-i

«-l Hlhr:

Fig. 7.

1 1

^ ^^-ip^;^ = ^ Lgii Fig. 8.

3.1. Suppose that A^ is not a k-th root of unity for k ^n. Then there is a unique element /^"^ G TLn such that (i) /(">e/ = 0 = e,- /(") /or 1 ^ / ^ n - 1. (ii) (/^"* — 1) belongs to the algebra generated by {e\,e2,..., e«-1}, (iii) f^"^f"^ = f^"^and (iv) An A^-A-^ LEMMA

PROOF. Uniqueness follows from (i) and (ii) as 1 — /^"^ is the identity of the sub-algebra generated by [e\,e2,.. .,en-\}. Further, (i) and (ii) imply (iii). Define /^^^ = 1, and /^^^ to be the empty diagram and Ao = l. Assume /^^^ has been defined fork^n. The identity of Figure 6 follows for / + 7 ^ « as /^^Ms 1 plus a sum of products of the ej. The latter terms die on contact with f^^^^\ Figure 7 shows an equality in TLn-\. The second diagram is in f^^~^^TLn-\ and so (by (i)) is a multiple of f^^~^\ The multiplier is determined by inserting both sides of the equation into copies of 5"^ and joining up the specified boundary points in the standard way. Suppose A"^ is not a /:-th root of unity for /: ^ « + 1. In Figure 8 is the definition of y(«+i)^ valid since Z\« ^ 0 by (iv). Properties (i) and (ii) for /("+^) then follow. All are immediate except for the property /^"+^^^„ = 0. However, multiplying the second diagram of Figure 8 by en produces a diagram containing a copy of the first diagram of Figure 7. That can then be excised at the cost of multiplying by An/An-i (use Figure 6). There results the right-hand side of Figure 8 multiplied by ^„ so f^^^^^en=0. Now place this defining equation for /^""^^^ into a disc with two holes (to be regarded as a projection of a genus 2 handlebody), joining the top two boundary points of the square around the left hole, and joining the other n points on the left to the other n on the right

Quantum invariants

715

Fig. 9.

by standard arcs that encircle both holes. The result is shown in Figure 9. In the first and third diagrams that should be clear. To form the second diagram, the left-hand square representing /^"^ in the second diagram of Figure 8 has been slid around near the boundary of the disc until it coalesces with the other copy of /^"^ (using /(")/(") = /^"^). If discs are added to the boundary components of this twice punctured disc to give 5^, this becomes, on dividing by An, the equation An+l + An-l = {-A-^

-

A^)An.

The required formula (iv) for Z\„+i follows from this at once by induction.

D

Now suppose the right-hand hole of Figure 9 is filled with a disc. One obtains an equality in the annulus and so in the skein space of 5*^ x D^. The first term is /("+^) inserted into the annulus with points on its left joined to those on its right by standard arcs around the annulus. That must represent some element in the skein space of S(S^ x D^) = C[Qf]. Let it be denoted Sn-\-i (oi). Then Figure 9, with help from Figure 7, gives Sn+iia)=aSn(a)

-

Sn-\(a).

Together with So(oi) = 1 and S\(a) = a, that is the defining recurrence relation for the Chebyshev polynomials of the second kind (renormalised). DEFINITION.

In the skein space of the solid torus, for each integer r ^ 3, define an element

cor by

r-2

Now the left-hand side of Figure 9 is symmetric with respect to the two holes except for the allocation of n and n — I. Thus summing, from « = 0 to n = r — 2, the difference between that identity and its analogue with the roles of the holes interchanged, produces the following. LEMMA 3.2. The difference between the skeins of Figure 10 can be expressed as the difference between two skeins, each containing a copy of f^^~^\

716

W.B.R. Lickorish

Fig. 10.

Note that in the above, cOr beside a framed curve really indicates the image of cOr under a framing preserving embedding of 5^ x D^ into the handlebody. The handlebody contains the other framed curve as shown. Now suppose that A is a primitive 4r-th root of unity. Suppose that a sphere separates S^ into two balls, one inside the sphere and the other outside it. Suppose that /^'""^^ G TLr-\ is placed in the inside ball and any element whatsoever, §, of TLr-\ in the outside ball (with framed points agreeing on the sphere). The resulting element of SS^ is zero. That is because ^ is a linear sum of products of the ei and of the element 1 e TLr-\. Of course f^^~^^ei = 0 and combining the/^'""'^ in the inner ball with 1 in the outer ball gives Ar-\. For A a 4r-th root of unity Ar-\ is zero. That gives impetus to the following definition. DEFINITION. TWO elements

of the skein space of a 3-manifold M are equal modulo f^^~^^ if their difference is in the subspace of elements that consist of f^^~^^ in a ball B in M together with any skein in M — B (with 2(r — 1) specified points on 95). The idea of skein space elements being equal modulo any other element of the skein of a ball, with boundary points, is defined in the same way. Theorem 3.1 reformulates the above remarks. THEOREM 3.1. The two elements of the skein of a genus two handlebody H shown in Figure 10 are equal modulo f^^~^\ Any inclusion of H into S^ induces a natural bilinear map

SH X S{S^ -H)^

SS^

and, when A is a primitive Ar-th root of unity, the two given elements produce the same element of the dual ofS(S^ — H). The curves labelled 1 in Figure 10 can be replaced by any collection of curves parallel in the given framing (apply the theorem to them one at a time) and then, using bilinearity, by linear sums of such parallels. In particular, the curves labelled 1, referred to in Theorem 3.1, can be replaced by curves labelled cor. (Technically, 'curves labelled cor' means images of COr under inclusions of solid tori respecting framings.) Suppose that L is a framed Hnk of n components temporarily oriented and ordered. There are canonical embeddings up to isotopy of S^ x D^ onto neighbourhoods of the components sending the natural product framing of S^ x D^ to the given framings. This induces a multilinear map (, , . . . , ) L : V S ( 5 ^ XD^)XS{S^

X D^) X • • • X S{S^

XD^)-^SS

3

Quantum invariants

111

Noting the similarity of Figures 10 and 1, the consequence of the above is this. 3.2. When A is a primitive Ar-th root of unity, the complex number {cor,o)r, ...,cOr)L does not change when L is changed by a K(2) move.

THEOREM

4. Invariants of 3-manifolds from skein theory The last result of the last section describes a way of associating a complex number to a framed link that is invariant under the second type of Kirby move on the link. It is now almost a matter of administration to produce an invariant for closed 3-manifolds. Suppose then that a 3-manifold M is obtained from S^ by surgery on a framed link L and let b^ and b- be the number of positive and the number of negative eigenvalues of A, the linking matrix for L. It is easy to check that those numbers do not change under a K(2) move. Let L^+ and U- be the unknot with framings ± 1. THEOREM

4.1. Let Abe a primitive Ar-th root of unity. The complex number

{0)r,(Or.

. . . , (Or)L{(^r)~u^

(<^r)^_"

is an invariant of M. That follows at once from Theorem 3.2 and the Kirby moves, though it is just necessary also to check (see below) that {cor)u^ {^r)u- # 0. Modulo possible re-normalisation, these invariants are, for r ^ 3, quantum SU^(2) invariants of M. The two identities of Figure 11 are exercises in the use of f^^\ The first diagram refers to n parallel strings containing /^"^ encircled by a parallel strings containing /^^^ (that is, encircled by a copy of Sa{(x)). Prove the result from the definition of /^"^ when a= I and recall the definition of the Chebyshev polynomials. The second formula is proved by induction on n. Suppose that A^^ ^ 1 and A^^^^ ^ 1; that is certainly the case when A is a primitive 4r-th root of unity. Consider the element x, say, of TLn shown in Figure 12 that consists of /^"^ encircled by oor. Modulo f^^~^\ this x is zero when n ^0 and is {cOr)u when n = 0 (here U is the 0-framed unknot). This is a most important fact which is proved in the following way. Note that -(A~^ + A^)x = -(A"^^"^^^ + A2^"+^^)JC, as is seen A

(-1)

A

PhA

=

- A

^ 2(«+l)

(-l)A

i\2n

Fig. 11.

^ -2(«+l)

- A

^ in)

-^

(„) /

718

W.B.R. Lickorish

hrDe

0

if n ;t 0

Fig. 12.

CDr

'<&' CD

iOr

r-l

E fI = 0

2 - 2 - 2

2

A

«

= .2r (A - A

)

-2

= /i

Fig. 13.

by introducing a small 0-framed unknot, slipping it over the component labelled cor (by Lemma 3.2) and then removing it using Figure 11 (when a = I). The result follows at once. That result together with the sliding property of cor shows, as in Figure 13, that {(^r)u+((^r)u- = {cOr)u whcu A is a primitive 4r-th root of unity. However, use of the definitions of cor and of An (and summation of a geometric progression) shows that r-2

{C0r)u=J2^^n = -H^^-^~^)

•

n=0

This is certainly non-zero and that tidies away the final detail of the proof of Theorem 4.1. A little calculation shows that 4r

{CO,)U,=J2A"'/{2A'^'\A'-A-% That, together with the formulae of Figure 11 gives enough information to calculate the invariants for lens spaces, Seifert fibrings over 5^ and, in fact, any 3-manifold with a surgery link consisting of A^ framed unknots simply linked together according to a pattern defined by a tree. A copy of cor must be placed on each of the A^ components, each cor must be expanded as Xl«=o ^nSnM and then the value in SS^ of each of the resulting (r — 1)^ links (with components labelled with the Sn{oi)) must be methodically calculated using Figure 11, starting from a 'leaf of the tree. There results a complicated summation. At least for lens spaces this has been extensively studied in [6,8,36], and [20] (watch for variants of notation and normalisation). It transpires that lens spaces Lp^q^ and Lp^^^ have the same SU^(2) invariants for all r if ^^ = ^ | = —1 mod/7 (for example, L65,8 and Lesjs). Further, calculation shows that the homotopy equivalent 3-manifolds L7 i and Lj^2 have distinct SU^(2) invariants.

Quantum invariants

719

It is easy to construct other pairs of 3-manifolds with all the same SU^ (2) invariants. Let X\ and X2 be the exteriors of two knots. Let h:dX\ -^ 9X2 be a homeomorphism and let —h be that homeomorphism composed with the automorphism of 9Xi that induces minus the identity map on the first homology group. Then manifolds X{Uh X2 and X\ U_/j X2 have all the same SU^(2) invariants [9,26]. The ideas outlined above have been extended in various ways. In particular if M has a spin structure, 6>, an invariant of (M, 0) can be defined so that the above invariant is, when r is even, the sum of these invariants over all spin structures. If M has a designated element of H\M; Z/2Z) then an invariant of the pair can also be defined. For details see [3,14] and [27]. An extension defining an invariant when A is a primitive 2r-th root of unity, r being odd, was made in [4]. It is often convenient to normalise these invariants in a way that is slightly different from that of the above theorem. If throughout cor is multiplied by a constant complex number /x, another invariant is produced that is the first invariant multiplied by /x^' where P\ is the first Betti number of the manifold M. Choose /x to be a real number such /x^ = (A^ — A~^)^/(—2r) then the normaUsed invariant IA(M) is defined by

where a A is the signature of the matrix of linking numbers of L. The first /x is a convention to achieve J^(5^ x 5^) = 1; it also gives XA{S^) = /x. LEMMA

4.1. If M is M with the other orientation then XA(M)

sums, iJiXA{M\ + M2)

= XA(M).

For connected

=XA{M{)XA(M2).

PROOF. If M comes from S^ by surgery on the framed link L, then M comes by surgery on L, the image of L under an orientation reversing homeomorphism of S^. Now A = A~^ and reflecting the crossing in Figure 2 exchanges the roles of A and A~^. Recall, too, that An is real and that Sn (01) is a polynomial with integer coefficients. Then the first result follows. The second result comes from the fact that the connected sum of two manifolds comes by surgery from the separated union of the framed links that give the individual manifolds. D

In the above, a closed connected 3-manifold M was obtained from S^ by surgery on an n-component framed link L with linking matrix A.lf M itself contains a framed link K, this K can be regarded as being in S^ — L and isotopies of K in M correspond to isotopies of ^ in 5^ — L together with 'Kirby-movements' of K across components of L. The embeddings of S^ x D^ onto neighbourhoods of the components of L now induce a multilinear map (, , . . . , ,K)L:S{S^

XD^)

XS{S^

X D^) X - - . X 5 ( 5 ^

XD^)-^SS^

by evaluating in SS^ the images of n skeins from S^ x D^ together with K.

720

W.B.R. Lickorish

THEOREM 4.2. Let A be a primitive Ar-th root of unity. An invariant of a framed link K in a closed orientable ?>-manifold M is given by

This induces a well-defined linear map IA(M,

):S(M)^C.

5. Some applications General skein theory considerations give immediately the following result. LEMMA 5.1. The map of framed links given by an embedding e:M\ -> M2 induces a map e: S{M\) -^ S{M2) which is linear if e is orientation preserving and semilinear if e is orientation reversing. Further, there is a natural identification S{Mx u M2) =S(Mx)®

S{M2).

The embedding of a disjoint union of solid tori onto the neighbourhood of a framed link has already illustrated both parts of this lemma. A second example is that, if F is any oriented surface, the embedding (F x I) u (F x I) ^^ (F x / ) , induced by including the two copies of the interval / onto the first and second halves of itself, induces an algebra structure on S(F x I). As a more extended example of this (and of Theorem 4.2) consider a handlebody H of genus g. The boundary of / / x [—1,1] is the connected sum of g copies of 5^ X S^ and so has invariant /x~^^~^\ There are natural disjoint embeddings of H onto H x {—1} and onto H X {1}. One, the first say, is orientation preserving the other orientation reversing. By the previous remarks, these two embeddings induce a map S(H) X S(H) -^ S{d{H X [ - 1 , 1])) which is a sesquilinear map, meaning that, as a function of two variables, it is linear in the first variable and semilinear (that is, conjugate linear) in the second variable. For A a primitive 4r-th root of unity, this map composed with XA{^{H X [—1,1]), ), becomes conjugate symmetric and so gives a Hermitian form on S{H). Let this form be denoted by 0 : S{H) X S{H) -^ C. This defines a map from S(H) to its dual space. Let 5(//)/ker be the quotient of S(H) by the kernel of this map. In fact, as will be indicated later, 5(//)/ker is a finite-dimensional vector space. Further, at least when A = exp(7r//2r), it can be shown that 0 is positive definite on 5(//)/ker and is hence a complex inner product on this quotient. A method for proving that will be outhned below, but for full details of this and related inner products see [48]. The boundary of a manifold always has a collar neighbourhood. Thus there is an embedding H u (dH x I) ^^ H which induces an action of S(dH x I) on S(H)/ktr as an algebra of endomorphisms. Suppose now

Quantum invariants

721

that e±:S^ x D^ ^^ dH x I SLTC two embeddings onto a neighbourhood of a simple closed curve C indH X ^ that send the natural framing of 5^ x D^ to (±1)-framings of the neighbourhood of C (the zero framing being the natural product framing). Then e±{ixcor) are endomorphisms of S{H)/kQv which will be denoted p ^ :S(H)/ker -^ «S(//)/ker. Note that p^ and p^ are mutually inverse endomorphisms; this follows from the identity of Figure 13 and the definition of fi. If x is any element of S(H), consider the image under the sesquilinear map of (p^x, p^x) in S(d(H x [—1, 1])). For this two copies of e^:S^ X D^ ^^ dH x I ^^ H have been used followed by inclusions into 9(// x [—1,1]). However the second inclusion is orientation reversing. Thus the second embedding can be isotoped out of / / X {1} through 8 / / x [ — 1,1 ] into / / x {—1} where it must be regarded as an instance of ^_. Then the combined effect of the two copies of iicor i n 5 ( 9 ( / / x [ — 1 , 1])) is that of p^p'^ which is the identity map. Thus p j is a unitary map with respect to the Hermitian form 0, namely 0(p^jc, p^x) = 0(x,x). Of course, any product of endomorphisms of the form p ^ is also unitary, because unitary automorphisms form a subgroup of the automorphisms of S{H). Suppose p is such a unitary automorphism. The Cauchy-Schwarz inequality implies for any x that |0(x, px)p ^ 0(J\C, x)0(px, px) and so | 0 ( X , P X ) | < |0(X,X)|.

5.1 (Originally in [7], see also [17] and [40]). Suppose M is a genus g closed orientable 3-manifold. If A = Qxpini/lr) then THEOREM

\IA(M)\^

2 . n ' - sm — r r

can be obtained by gluing together two copies of a genus g handlebody H; so can the connected sum of g copies of S^ x S^. The gluing map that gives this latter connected sum can be changed by a product of twist homeomorphisms of dH so that the new gluing gives M (see [21]). Thus M can be obtained from d(H x [—1,1]) by ± 1 surgery on simple closed curves each in some dH x t. Thus a surgery link L, in 5^, for M is a surgery link for d(H x [—1, 1]) (for example, the 0-framed unlink of g components) together with these extra surgery curves. Now \XA(M)\ = \/ji{ijicor, ficor,..., /xa>r)Ll for {^icOr)u- has unit modulus. The right-hand side of this is, in the above notation, |0(0, p0)| where 0 denotes the empty skein in H and p is the composition of the p ^ where C runs through the above mentioned curves in the dH x t. However |0(0, 0)| is just | X A ( 9 ( ^ X [—l,l]))| = /x~^^~^\ Hence the above instance of the Cauchy-Schwarz inequality implies PROOF. M

that \IA{M)\ follows.

< |/x~^^~*^|. But A = Qxp(7ti/2r) implies that lJi = Jj: sin f and the result D

This result, which gives a lower bound for the Heegaard genus of a 3-manifold, is at least one place where the quantum invariants interact with the concerns of classical 3-manifold theory. Y. Yokota develops these ideas in [48] where he considers skein spaces of a handlebody with some specified framed points on its boundary. The same general ideas concerning Hermitian forms lead him to the following generalisation of the above.

722

W.B.R. Lickorish

THEOREM 5.2. Suppose a framed knot K in a closed orientable ?>-manifold M meets each half of a genus g Heegaard splitting ofM in n standard {boundary parallel) arcs. Let A = exp(7r//2r) then

\lA{M,K{Sk(a)))\^

[2

n

\-g

J - sin —

V r

r

sini^±^ sin^

Here TA{^, K(Sk(oi))) is the invariant of M including in it the skein Skicc) about the knot K. A sharper inequaUty is available, [48], when M is a homology 3-sphere. Similarly these ideas can be interpreted in terms of the tunnel number of a knot in a 3-manifold (that is the minimal number of arcs that need to be added to the knot to produce a spine of one side of a Heegaard splitting) in the following way (see [48] and [17]). THEOREM 5.3. If K is a knot with tunnel number N in a closed oriented 3-manifold M and A = exp(7r//2r) then r-2

^|jA(M,/^(5^(a)))|^^ k=0

2 . 71 - sin — r r

-2N

Again, a sharper result is available for a homology 3-sphere; the right-hand side can be multiplied by 2~^. The techniques just described were adapted in [35] to prove, for the first time, that certain Montesinos knots have tunnel number two. Suppose a is an arc added to a knot K in S^, so that Of U ^ is a spine of a handlebody H that is one side of a Heegaard splitting of S^. Then there is an involution L of S^, leaving K invariant and having fixed point set an unknot U that meets K in two points and H in three intervals. Then S^/L is also a copy of S^ with H/L a ball. (K U U)/L is a, probably knotted, ^-curve meeting dH/i in six points. It is the skein theory of the ball with six sets of boundary points (each labelled with an idempotent of a Temperley-Lieb algebra) that is considered. Pairing this with the similar space for the complementary ball gives a Hermitian form and (with some care with the labellings) elements of the six string braid group give unitary automorphisms. The resulting CauchySchwarz inequality concerns the evaluation in SS^ of the above ^-curve with labelled edges and triads at the vertices. However, symmetries of Montesinos knots are classified so this can be checked against evaluations on the only possible ^-curves that can arise. A contradiction sometimes results. Although this proof uses all the technology of triads and idempotents, it does not really use the 3-manifold invariants, for all the calculations are in S^ and cor is never mentioned. Towards the end of [14], Kirby and Melvin study these SU^ (2)-invariants, when A is a primitive 4r-th root of unity, for small values of r. They make use of the fact that the Jones polynomial at t = e^^'/^ is, for certain small values of r, expressible in terms of more classical invariants [30]. They also employ a Symmetry Principle (see also [25]) which has the effect of reducing the number of terms in the expansion of cor. When r = 3, TA(M) is expressed in terms of the Brown invariant of the linking matrix of a surgery link and of the dimension of H\M; Z/2Z). When r = 4, J^ (M) is simply expressed in terms of the

Quantum invariants

723

/x-invariants of all the spin structures on M. When r = 5 there is, when M is a homology 3-sphere gained by Dehn surgery on a knot, a correlation between XA (M) and the Casson invariant of M modulo 5. The following result, also, appears in [14]. 5.4. Suppose the Jones polynomials of knots K\ and K2 have distinct evaluations at e^^'/^. Then, for every integer n, surgery on S^ along K\ and K2, each with framing n, gives distinct oriented 3-manifolds. THEOREM

In the preamble to Theorem 5.1, with A a primitive 4r-th root of unity, it was shown how a twist homeomorphism about a simple closed curve C in the boundary of a handlebody H led to automorphisms p ^ of <S(//)/ker. This induces a projective representation of the mapping class group of H on S(H)/kcr (see [40]). It is only projective as powers of the unit modulus complex number {ii(jOr)u- have to be 'overlooked'. That however is rectified in [33] where a genuine representation, of an extension by Z of the mapping class group, is obtained essentially in this way. In [16] 3-manifold invariants obtained by means of representations of mapping class groups are discussed. Further, it is shown in [5] that <S(//)/ker is the vector space that should be associated with the surface 9// in a Topological Quantum Field Theory that is associated with the theory of this SU^ (2) invariant of closed 3-manifolds. Roughly, the idea is that a cobordism between dH\ and 9//2 can be obtained by surgery on a framed link in Hj that lies in the complement of a copy of H\. Placing cor about each surgery curve and an arbitrary skein in the copy of H\ should lead to an element of S(H2). In this way the cobordism might lead to a linear map S(H\ )/ker -^ «S(//2)/ker, that being the essence of a Topological Quantum Field Theory. There are however, difficulties with the term in {/jicor) u_ that occurs in the definition of J^ (M) that necessitate the supposition that each cobordism be equipped with a Pontryagin class (or 'given a /^i -structure'). The details are fully explained in [5]. 6. Fusion To proceed further one must use a little more daring with the Temperley-Lieb algebras and their idempotents than has so far been attempted here. Although some progress can be made without any quotients, [27], it will usually be appropriate to work with skeins modulo/(''-'^ The first diagram of Figure 14 shows a 'triad' element of the skein space of a ball B with a -\-b -\- c framed points specified on its boundary. The diagram will be abbreviated, to a black dot with three labelled protruding edges, as shown. Note the presence of the idempotents of the various TLn and note that if a = 0 then b = c and the triad is just f^^\ If B c S^ then the closure of S^ — B is also a ball and a base of the skein of this outside ball is (projecting onto a disc) represented by diagrams of arcs with no crossing. Combining such a base element with the triad will always give the zero element of SS^ except possibly when a similar triad is placed in the outside ball and there results the element 0(a,b,c) SLS shown. A httle work [28,34] and [11], shows that ^(«,Z7,c) = (Z\^+,+,!Z\;,_,!Z^v-i!^z-i!)/(^v+z-i!^z+.-i!^x+v-iO,

724

W.B.R. Lickorish a

b^

= eia,b,c) Fig. 14.

Fig. 15.

A base is all

:M: Fig. 16.

where An! denotes Z\„ zl„_i Z\„_2 ... ^ i, this being 1 if n is — 1 or zero. It follows, when A is a primitive 4r-th root of unity, that it is precisely when (a, b, c) is r-admissible in the following sense that the triad exists and 0(a,b,c)^0. DEFINITION. The triple of non-negative integers (a, b, c) is r-admissible provided that a -\-b-\-c is even, a^b-\-c,b^c-\-a,c^a-\-b and a-\-b-{-c ^ 2{r — 2).

All black dots in the following diagrams will refer to r-admissible triples, and the phrase 'modulo r' will mean 'modulo r-inadmissible triples' (which include /^^~^^). The identity of Figure 15 is an easy exercise. The left-hand diagram of Figure 16 symbolises the subspace of the skein space of the ball, with a-\-b-\-c framed boundary points, that has idempotents as shown and any skein element in the middle smaller ball. That subspace has dimension one or zero, according as to whether an {a, b, c) triad exists. Modulo r it has dimension one if and only if (a, b, c) is r-admissible. The right-hand diagram similarly refers to the ball with a-\-b-\-c-\-d framed boundary points. The subspace here has, modulo r, a base of all r-admissible elements as shown. To show these elements span employ the fact that, in TLr-\, 1 = /^'^"^^ + £" where £• is a sum of products of the ei. Here f^^~^^ can be forgotten, so a link with r — 1 parallel arcs travelling from right to left can be replaced with a sum of links with fewer arcs going all the way from right to left. Independence follows by adjoining a triad to a postulated linear dependence. Details are in [28,34] and [11]. This base has a 'horizontal bias' so there is another base with a 'vertical bias'. The change of base formula is in Figure 17, the summation being over all / for which both the triads in the diagram are r-admissible. The terms of the change of base matrix {?^)} are

725

Quantum invariants

^b

c

r,^af}9{adk)9(bck)A\'

^Tf\

Fig. 17.

a+b-c ^ ^ . ^ , fl>^> -c2 ^— fl+6-c+ 2 c

«^

(-1) ' A

< : > < :

Fig. 18.

admissible

Fig. 19.

:t^^")o(•

1

e (a, b, c)

Fig. 20.

traditionally called '67-symbols'. Adjoining triads to the change of base formula easily produces the expression shown for a 67-symbol, in terms of a framed, labelled, tetrahedral graph with triads at the vertices, regarded as an element of SS^ = C. In theory one knows these 67-symbols. A lengthy, closed formula is known for them [3, 11,15]. The equality of Figure 18 is, however, a routine exercise. Some expression of the form of Figure 19 must be true. The coefficients come easily by adjoining triads. In principle, the formulae of Figures 15, 17-19, together, of course, with knowledge of formulae for An, 0(a,b, c) and (regrettably) the 67-symbols, give all that is required to calculate, modulo f^^~^\ the value in SS^ of any framed link with components decorated with various 5'n(Qf)'s. Then XA{M) is a linear sum of such terms. Use Figure 19 to fuse strands near crossings. Figure 18 to remove the crossings. Figure 17 to change trivalent labelled framed graphs until they contain 2-gons, then remove 2-gons using Figure 15. There results an awesome sum of products of the An,0(a,b, c) and 67-symbols. The formulae of Figure 20, when A is a primitive 4r-th root of unity, follow easily from Figures 19 and 12. The right-hand expression is to be taken to be zero unless (a, b, c) is r-admissible.

726

W.B.R. Lickorish

Fig. 21.

^2g + l

"2g + 2

Fig. 22.

Suppose one applies Figure 19 and then Figure 15 to two zero framed curves encircling S^ X D^ labelled with Sa (a) and Stiot). Then one obtains the following (Clebsch-Gordan) formula in 5(5"^ x D^) modulo r. Sa{a)Sb(a) =

^Sc{a) c

the sum being over all c such that (a, b,c) is r-admissible. Application of Figure 20 to the element of S(S^ x D^) shown in Figure 21 shows that element to be /x"^ Xl^^o^'^^^^W^- Suppose g copies of Figure 21 are threaded (like beads) on a 0-framed 6^^-decorated unknot in S^. The resulting element of S(S^), modulo r, is, using the above Clebsch-Gordan formula and Figure 12, /x~^^^+^^A^g where A^^ is the number of ways that the Clebsch-Gordan formula produces an So(a) term from This A^^ is just the number of ways of labelling the edges of the graph of Figure 22 with non-negative integers {at} so that an r-admissible triad appears at each vertex. THEOREM 6.1. Let F be a closed orientable surface of genus g and let Abe a primitive Ar-th root of unity. Then XA{S^ X F) = Ng

PROOF. A surgery diagram for 5^ x F is the sum of g copies of the 0-framed Borromean rings (with one component featuring in every summation). It has just been shown that, in SS^, this framed link, with every component decorated with cor, is /x~^^<^"^^^A^g. To obtain 1A{S^ X F) this must be multiphedby a /x for each link component and one more for convention. (Note that the framed link has zero linking matrix which has zero signature.) The result follows. D This result means, for ^ = 1, that XA{S^ X S^ x 5 ^ ) = (r — 1 ) . A similar calculation applied to the link of Figure 23 shows that the manifold gained from it by surgery has the

Quantum invariants

727

Fig. 23.

Y ? 9

y

Fig. 24.

same invariant. That manifold is the (S^ x 5^)-bundle over S^ that has minus the identity as monodromy map. The first homology group distinguishes this from S^ x S^ x SK Now regard the graph of Figure 22 as the spine of a genus g handlebody H. Each labelling of the edges of the graph, with r-admissible triples at vertices, gives, on inserting triads at the vertices and connecting them all up with parallel arcs along the edges, an element of SH. The set of elements coming from all such labellings spans SH modulo r. This follows by a similar argument to that outlined above for the ball with a-^ b -\-c -\-d points in its boundary (see Figure 16). There are two copies of this graph in Figure 24. Inserting skeins into the neighbourhoods of the two graphs and evaluating in SS^ gives a pairing SH x SH ^^ C. When A is a primitive 4r-th root of unity, this pairing is an orthogonal pairing on the set of skeins given by r-admissible labellings of the graphs. That can, at once, be used to show that skeins coming from distinct such labellings are independent and hence form a base (more details are in [28] and [34]). Thus A^^ is the dimension of SH modulo r. If the handlebody neighbourhood of just one of the graphs is given an orientation disagreeing with that of S^, this pairing is the same as the pairing 0 : S(H)/ker x S(H)/kcr -^ C already considered for Theorem 5.1. Now using the given base elements it is straightforward, using Figure 20, to make calculations (as required in Theorem 5.1) on Figure 25 to determine when 0 is positive definite.

7. The state sum invariants of l\iraev and Viro Other invariants of a closed orientable 3-manifold M, closely allied to the invariants just discussed, are the Turaev-Viro invariants [44]. Many calculations of these invariants can be found in [11]. They were defined as state sums, as sums over all possible 'states' of the products of some 'weights' associated to the state. Invariance was checked either by using the Matveev moves on a special spine of the 3-manifold, or by using the Pachner

728

W.B.R. Lickorish

moves [38] on a triangulation. Later Turaev proved that this invariant is the square of the modulus of a corresponding SU^ (2)-invariant. Though in one way that is disappointing, in that the Turaev-Viro invariant is therefore a weaker invariant, the greater calculabiUty of a state sum might lead to applications. This has all been re-cast into a very simple form, by Roberts [41], in the following way. Consider first a genus g Heegaard splitting diagram for M. This is taken to be two disjoint sets of curves in a handlebody H\ a complete set in 9// of meridian curves {5/} for / / , and a complete set of meridian curves {si} for the closure of M — H pushed into H along a collar ofdH. All the curves inherit framings from dH. The framed link L(s) in S^, corresponding to this splitting diagram, is that consisting of the images of these curves under a standard embedding of H into S^. (Actually, any embedding will do, see [41].) The signature of the linking matrix of L(5) is zero as the matrix contains a g x g block of zeros. Further, L(s) is a surgery diagram for M + M. This can be seen by the Kirby-calculus technique of regarding the surgery process as giving a 3-manifold as the the boundary of a 4-ball with added handles; regard the 5-curves as giving 1-handles and the 6:-curves as being attaching circles for 2-handles [13]. It follows from Lemma 4.1 that IJA(M)|

= fl^{fXO)r,llO)r,

. . . , fJ^COr)Lis)-

Now consider the 3-manifold M to be triangulated with df /-simplexes. A regular neighbourhood of the dual 1-skeleton is a handlebody H. Take on dH a. curve 5/ that is the intersection of dH with the i-ih 2-simplex. Let £i be the result of pushing a little into H a curve in dH that bounds a disc in M — H which intersects transversely the 1-skeleton at the barycentre of the /-th 1-simplex. Framings on the curves come from dH. Now regard H as embedded in some standard way in S^ and let Lit) be the framed link in S^ of all the s and 5 curves coming from the triangulation in the above way. This is not quite the link from a Heegaard splitting; some of its curves must be deleted, namely the d^, — \ 5-curves corresponding to a maximal tree of the dual 1-skeleton and the d^ — \ e-curves corresponding to a maximal tree of the 1-skeleton. Let L{t)~ be the link after deletions. If all components of L{t) are decorated with ^icor those corresponding to curves not in L{t)~ can be slid over curves of L(t)~, into small balls, without changing the resulting element of ^^-^ Thus \lA{M)f

= /0+^MM6Or, /X6^r, . . . , /X60r)L(r).

The link L{t) is Roberts' 'chain mail' link, the reference being to mediaeval armour. Certainly, if one constructs this link from a fine subdivision of a given triangulation it is a link that occupies (protects) the manifold in a ubiquitous manner; operations on it might be reminiscent of integration. Notice that in the link L{t) a (5-curve bounds a disc in H (namely H intersected with a 2-simplex) that meets three and only three e-curves. To evaluate {jicor, jjicor,..., fJicor)L{t) consider the multilinear expansion that occurs on substituting cor = Yln^o^nSniot) for each e-curve but leaving cOr 'intact' on the 5-curves. Then use the right-hand identity of Figure 20 on each labelled diagram of the resulting summation (recall that, in Figure 20, each side of the identity is non-zero only when (a, b, c) is r-admissible). Each

Quantum invariants

729

labelled diagram then breaks up into many labelled tetrahedral graphs and 0-framed unknots labelled cor • Tidying up the details, Roberts obtains the following theorem. The state summation expression here given for \XA(M)\^ is, up to normalisation conventions, the Turaev-Viro invariant of M. THEOREM 7.1. Let M be a closed orientable 3-manifold with a triangulation having do 0-simplexes. A state is a function s from the set of edges of the triangulation to {0, 1, 2 , . . . , r — 2} that maps the three edges of any 2-simplex to an r-admissible triple. Then, if A is a primitive Ar-th root of unity,

s

^ e

f

t

The summation is over all states, e, f and t are the edges, faces and tetrahedra of the triangulation, s maps the edges of face f to the triple sf. Similarly T(st) is the skein evaluation of a tetrahedral graph, with triads at the vertices and edges labelled with the values of s on the edges of the tetrahedron t {see Figure 17).

8. Invariants from HOMFLY-based skeins Quantum 3-manifold invariants corresponding to SU(A^) fox N >2 have been substantiated by Turaev and Wenzl [45] using intricacies of representation theory. More recently a more geometric skein theory proof of the existence of these invariants has been developed by Yokota [49]. The proof is self-contained though clearly inspired by representation theory. An outline of it now follows. Throughout, A^ will be a fixed integer, N ^2 and t eC a. primitive 2N(K -h A^) root of unity for some 'level' K. Let M be an oriented 3-manifold together with a finite set of framed points in 3M, each such point being oriented (that is, it has assigned to it a direction 'into' or 'out of the manifold). DEFINITION. The linear skein SM(M) is the C-vector space of formal linear sums of framed oriented links in M, that consist of closed curves and arcs that meet 9M in precisely the given framed oriented points, quotiented by: (i) isotopy of framed links; (ii) L U O = [A^]^, where [k] = (t^^ - t-^^)/(t^ - 1 ' ^ ) ; (iii) adding ±1 to the framing of any component of L changes L to r^^^ ~'^L; (iv) the skein identity of Figure 25.

The familiar theory of the HOMFLY polynomial, [31] for example, asserts that again the skein space of S^ is one-dimensional and that, if the 0-framed unknot is taken as its

Fig. 25.

730

W.B.R. Lickorish

base, the coordinate of any 0-framed knot L is the HOMFLY polynomial of L with the two variables taken to be i^^ and i(^~^ — ^^), in the notation of [31]. This is well known to be the substitution corresponding to SU(A^). Nevertheless, in what follows the empty diagram is taken as base for SN(S^) and is used to identify it with C. The skein space of the cube with / in-going points on its bottom face and / out-going points on its top face will be SN(BJ). If ^,r] e SjsjiBJ), placing § above r] induces a well defined product and S^iBJ) becomes an (Hecke) algebra. If ^ G SM{B^) and rj e S^BJ then juxtaposing cubes side by side produces ^^r] eS^ (B^^/). Similarly a tensor product m comes by embedding two solid tori in one. If § G SN(BI) then $ denotes the element of SN(S^ X D^) obtained by placing the cube containing § in the solid torus in a standard way and joining the / points at the top of the cube to those at the bottom of the cube by / embedded arcs each encircling the solid torus in a positive direction. The association § i-^ § induces a linear map Sis/(BJ) -> Sjsi(S^ x D^). Another linear map X''^N(S^ X D^) -^ C comes from taking an embedding, preserving orientation and framing, of S^ x D^ onto a neighbourhood of a 0-framed unknot. Suppose that cr/ is the usual i-ih generator of the /-string braid group which provides diagrams for elements of Sjs/iBJ). Regarding a, as an element of SMiBJ), it is clear that {or.^^: 1 < / < / — 1} is a set of generators for SM(BJ) as an algebra. Yokota [49] focusses attention on two special elements of S^^iBJ). Both are idempotents. The first f^^\ the symmetriser, has the property that for each /, ^±1 y(/) ^ ^±(yv-l)y(/) ^ y(/)^±l ^ the anti-symmetriser g^^^ is such that a ± ' / ) = -,^(A'+')^(/)^g(/)^±.. These are combined together to form further idempotents corresponding to Young diagrams. A Young diagram X is a way of partitioning a finite set of |A| elements, represented by little squares, in two related ways; the sets of the partitions are the rows and columns of a diagram of little squares as in Figure 26. The Young diagram A is specified by a sequence [/A^-I , IN-2^..., /2, /i ] of non-negative integers where there are /A^_I + IM~2 H h // squares in the i-ih row (the top row being the first); there are A^ — 1 rows though the final rows may be empty. The idempotent ex G S!\f(Bl!) is a scalar multiple of the element formed in the following way. Take an anti-

Fig. 26.

Quantum

731

invariants

H)^ Fig. 27.

symmetriser of the size of each column of the diagram and place them side by side to form (l)\<^/i

>{n connect this to the element f(lN-i)

(g) y(/yv-i+/yv-2) ( g ) . . . (g) ^ (/^_,+/^_2 + - + / l )

formed by juxtaposing symmetrisers with sizes the rows of the diagram, and connect that to a second tensor product of anti-symmetrisers, identical to the one first mentioned, in the way shown in Figure 27 (where the g-idempotents are black rectangles and the /-idempotents are white ones). The complex number Axis defined by

In [49] it is, at some inconvenience, shown that N-\

^^= n

n

Y\\ln+ln-\+--'

+ li+n-i

+ \]/[n-i

+ \l

Now, for K a fixed positive integer the set of Young diagrams F^^K is defined by

rN,K = h'

Y,ii^K\ /=i

and the element QK eS^iS^ QK=

^

x D^) is defined by

A{ex-

Recall that f is a primitive 2N(N + K) root of unity. The main result of Yokota [49] is that, if QK occurs as the decoration on any component C of a framed Hnk in S^, then that link may be changed by pushing (according to the fashion of a Kirby K(2) move) any other component with either orientation across C, and (retaining the ^K decoration on C) the

732

W.B.R. Lickorish

evaluation of the decorated link in SN{S^) is not altered. Thus in this situation QK has exactly the same property as had coy in the previous discussion. Then, in the same way as before, an SU^ (A^) invariant for an oriented 3-manifold M is constructed as is stated in the next theorem (from [49]). 8.1. Suppose the 3-manifold M is obtained by surgery on a framed link L in S^. A well-defined invariant XM,K(^) is defined by

THEOREM

Here { , , . . . , }L is the multilinear map on 3^(5^ x D^) coming from L, U- is the unknot with framing — 1, the signature of the linking matrix of L is a and ^ is a normalising factor defined so that 0~^ = {^K)U where U is the 0-framed unknot. As before

and that is non-zero. Yokota observes that if Q"^ is defined to be the same as QK but with every string orientation in its construction reversed, then Q"^ may be used in place of any (or every) occurrence of ^A: , without changing the outcome in any way. That is because Q"^ will still have the fundamental property that, in any decorated framed link in 5^, strings maybe slid over ^ ^ without changing evaluations in SN{S^)- SO in S^iS^ x D^), modulo the effect of the root of unity, ^^QK = (^K)U^K ~ {^'^)U^K by performing each of the permitted slidings. However changing all the orientations of a link never changes its HOMFLY polynomial. Thus {QK)U = {^'K)U and, modulo the root of unity, Q'j^ = QKClearly, investigation of the SU<^ (A^) invariants can be attempted along the lines used above for the SU^ (2) invariants. However the necessary coefficients are, at every point, distinctly more cumbersome. In a way similar to that of Theorem 6.1 it can be shown (see [29]) that 1N^K{S'^S'

^S')

= \r^^Kl

Similarly, for the {S^ x 5^)-bundle over S^ obtained by surgery on Figure 23, TN,K{S^

X (5^ X S^)) = \[X e F^^K:

A=

r}|.

Here X* is the Young diagram dual to A; if X is specified by [/A^-I , IN-2, ---Ji] then X* is specified by [/1, /2,. •., /A^- i ]. When A^ = 2 every A, is self-dual but otherwise that is not so. Thus these manifolds are distinguished by their SU^(A^) invariants for A^ > 2 but not by their SU^ (2) invariants. For lens spaces calculations can be made in the same way as before but the results are not immediately attractive; some of the necessary formulae for doing this, obtained from a perspective different from skein theory, can be found in [18].

Quantum invariants

733

9. Conclusion This article has attempted to give some details of some of the simplest aspects of the theory of quantum invariants. The result is but a scratching of the surface of a vibrant, far-reaching, rapidly developing subject that has gained a momentum of its own. Because this topic interacts with many disciplines it has been interpreted in many different ways with respect to varying agenda. Recently there has been considerable investigation into the behaviour of the SU^ (2) invariants as r (in the above notation) tends to infinity; that is Witten's 'large k limit'. An interpretation by Ohtsuki [37] led to a formulation of 'finite type' 3-manifold invariants, which are an analogue of the Vassiliev invariants in knot theory [2]. A survey of that topic is in [32]. The analogue is followed further in [19] where a universal 3-manifold invariant is constructed. This gives, for a 3-manifold, a class of formal sums of trivalent graphs to which the combinatorics of a Lie algebra may be applied to obtain a numerical invariant; again, surgery and the Kirby calculus provide the contact with actual 3-manifolds and the universal Vassiliev-Kontsevich invariant for knots is the inspiration. In all of the quantum invariant theory it is remarkable that Witten's original formula has led to so much creativity and there is the feeling that there are more insights to come.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17]

M.F. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press (1990). D. Bar-Natan, On the Vassiliev knot invariants. Topology 34 (1995), 423^72. C. Blanchet, Invariants on three-manifolds with spin structure. Comment. Math. Helv. 67 (1992), 406-427. C. Blanchet, N. Habegger, G. Masbaum and P. Vogel, Three-manifold invariants derived from the Kaujfman bracket. Topology 31 (1992), 685-699. C. Blanchet, N. Habegger, G. Masbaum and P. Vogel, Topological quantum field theories derived from the Kauffman bracket. Topology 34 (1995), 883-927. D.S. Freed and R.E. Gompf, Computer calculations of Witten's 3-manifold invariants, Comm. Math. Phys. 141(1991), 79-117. S. Garoufalidis, PhD dissertation. University of Chicago (1992). L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semi-classical approximation, Comm. Math. Phys. 147 (1992), 563-604. J. Kania-Bartoszynska, Examples of different 3-manifolds with the same invariants of Witten and Reshetikhin, Topology 32 (1993), 47-54. C. Kassel, Quantum Groups, Springer (1995). L.H. Kauffman and S. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Ann. of Math. Stud., Vol. 134, Princeton University Press, Princeton, NJ (1994). R.C. Kirby, A calculus for framed links in S^, Invent. Math. 45 (1978), 35-56. R.C. Kirby, The Topology ofA-Manifolds, Lecture Notes in Math., Vol. 1374, Springer (1989). R.C. Kirby and P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C), Invent. Math. 105 (1991), 473-545. A.N. Kirillov and N.Y. Reshetildiin, Representations of the algebra Uq{S\{2)), q-orthogonal polynomials and invariants of links. Infinite Dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys., Vol. 7, World Scientific (1989), 285-339. T. Kohno, Topological invariants of three-manifolds using representations of mapping class groups I, Topology 31 (1992), 203-230. T. Kohno, Tunnel numbers of knots and Jones-Witten invariants. Braid groups, knot theory and statistical mechanics II, Adv. Ser. Math. Phys., Vol. 17, World Scientific (1994), 275-293.

734

W.B.R. Lickorish

[18] T. Kohno and T. Takata, Symmetry ofWitten's 3-manifold invariants for sl(/i, C), J. Knot Theory Ramifications 2 (1993), 149-169. [19] T.T.Q. Le, J. Murakami and T. Ohtsuki, On a universal perturhative invariant of 3-manifolds. Topology 37 (1998), 539-574. [20] B.-H. Li and T.-J. Li, Generalized Gaussian sums and Chern-Simons-Witten-Jones invariants of lens spaces, J. Knot Theory Ramifications 5 (1996), 183-224. [21] W.B.R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 (1962), 531-540. [22] W.B.R. Lickorish, Surgery on knots, Proc. Amer. Math. Soc. 60 (1977), 296-298. [23] W.B.R. Lickorish, Invariants for 3-manifolds from the combinatorics of the Jones polynomial. Pacific J. Math. 149 (1991), 337-347. [24] W.B.R. Lickorish, Three-manifolds and the Temperley-Lieb algebra. Math. Ann. 290 (1991), 657-670. [25] W.B.R. Lickorish, Calculations with the Temperley-Lieb algebra. Comment. Math. Helv. 67 (1992), 571591. [26] W.B.R. Lickorish, Distinct 3-manifolds with all SU(2)^ invariants the same, Proc. Amer. Math. Soc. 117 (1993), 285-292. [27] W.B.R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), 171-194. [28] W.B.R. Lickorish, Skeins and handlebodies. Pacific J. Math. 159 (1993), 337-349. [29] W.B.R. Lickorish, Sampling the SU(A^) invariants of three-manifolds, J. Knot Theory Ramifications 6 (1997), 45-60. [30] W.B.R. Lickorish and K.C. Millett, Some evaluations of link polynomials. Comment. Math. Helv. 61 (1986), 349-359. [31] W.B.R. Lickorish and K.C. MiWtii, A polynomial invariant of oriented links. Topology 26 (1987), 107-141. [32] X.-S. Lin, Finite type invariants of integral homology 3-spheres: a survey, Banach Centre Publ., Vol. 42, PoHsh Acad. Sci., Warsaw (1998), 205-220. [33] G. Masbaum and J.D. Roberts, On central extensions of mapping class groups. Math. Ann. 302 (1995), 131-150. [34] G. Masbaum and P Vogel, 3-valent graphs and the Kauffman bracket. Pacific J. Math. 164 (1994), 361-381. [35] K. Morimoto, M. Sakuma and Y. Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48 (1996), 667-688. [36] J.R. Neil, Combinatorial calculations of the various normalisations of the Witten invariants for 3-manifolds, J. Knot Theory Ramifications 1 (1992), 407-499. [37] T. Ohtsuki, A polynomial invariant of integral homology 3-spheres, Math. Proc. Cambridge Philos. Soc. 117(1995). [38] U. Pachner, P.L homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combinatorics 12 (1991), 129-145. [39] N.Y. Reshetildiin and V.G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), 547-597. [40] J.D. Roberts, Skeins and mapping class groups. Math. Proc. Cambridge Philos. Soc. 115 (1994), 53-77. [41] J.D. Roberts, Skein theory and Turaev-Viro invariants. Topology 34 (1995), 11\-1%1. [42] S. Sawin, Links, quantum groups and TQFTs, Bull. Amer. Math. Soc. 33 (1996), 413-445. [43] V.G. Turaev, Quantum Invariants for Knots and 3-Manifolds, de Gruyter, Berlin (1994). [44] V.G. Turaev and O.Y. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31 (1992), 865-902. [45] V.G. Turaev and H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, Intemat. J. Math. 4 (1993), 323-358. [46] A.H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503-528. [47] H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. IX (1987), 5-9. [48] Y. Yokota, On quantum SU(2) invariants and generalised bridge numbers of knots. Math. Proc. Cambridge Philos. Soc. 117 (1995), 545-557. [49] Y. Yokota, Skeins and quantum SU(N) invariants of 3-manifolds, Math. Ann. 307 (1997), 109-138.

CHAPTER 15

L^-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes Wolfgang Liick Fachbereich fiir Mathematik und Informatik, Westfdlische Wilhelms-Universitdt Munster, Einsteinstr. 62, 48149 Munster, Germany E-mail: lueck@math. uni-muenster.de

Contents Introduction 1. L^-Betti numbers for CW-complexes of finite type 2. Basic conjectures 3. Low-dimensional manifolds 4. Aspherical manifolds and amenability 5. Approximating L^-Betti numbers by ordinary Betti numbers 6. L^-Betti numbers and groups 7. Kahler hyperbolic manifolds 8. Novikov-Shubin invariants 9. L^-torsion 10. Algebraic dimension theory of finite von Neumann algebras 11. The zero-in-the-spectrum Conjecture 12. Miscellaneous References

HANDBOOK OF GEOMETRIC TOPOLOGY Edited by R.J. Daverman and R.B. Sher © 2002 Elsevier Science B.V. All rights reserved

735

737 738 745 749 753 757 759 764 770 778 798 803 808 809

This Page Intentionally Left Blank

L -invariants of regular coverings of compact manifolds and CW-complexes

737

Introduction When we were asked to write this survey article for the Handbook of Geometry, we began to wonder who will read such an article. Well, certainly not someone who wants to fix his car. The possible readers could be (1) students or more advanced mathematicians who are looking for new problems, (2) people who are curious about this topic and want to get a first impression, (3) others who are interested in this topic already and want to invest some time to learn more about the techniques, (4) experts who want to see new developments and an extensive list of literature (with all their papers cited) or (5) people who want to read and absorb something new without big effort while watching a Star Trek episode or a soccer game. We hope that any of these groups can get something from this survey article. The appeal of the topic comes from its connections to rather different fields of mathematics like spectral theory of the Laplace operator, questions about metrics with certain curvature properties on manifolds, isomorphism conjectures in algebraic A'-theory and the BaumConnes Conjecture, low-dimensional manifolds, group theory, index theory, intersection homology, representation theory of Lie groups and so on. For the groups (1) and (2) we reconmiend to glance at Section 1 and then read Section 2 where the main problems are stated. Moreover, there are other conjectures and open problems stated throughout the other sections. We treat Conjecture 2.1 about the rationality of the L^-Betti numbers, the Singer Conjecture 2.6 about the vanishing of the L^-Betti numbers outside the middle dimension of the universal covering of an aspherical closed manifold, the Hopf Conjecture 2.7 which is a version of the Singer Conjecture for closed Riemannian manifolds with negative sectional curvature. Conjecture 8.9 about the rationality and positivity of the Novikov-Shubin invariants. Conjecture 9.10 about the triviality of the homomorphism 0r : W h ( r ) -^ R^^ given by the Fuglede-Kadison determinant. Conjecture 9.18 about the relation of the simplicial volume and the L^-torsion of aspherical closed manifolds. Conjecture 9.24 about the vanishing of the L^-torsion for closed aspherical manifolds with amenable fundamental groups and the zero-in-the-spectrum Conjecture 11.1 and 11.4. None of these problems seem to be easy and have created a lot of work as discussed in Sections 3, 4, 7, 8 and 9. These sections and Sections 5, 6, 10 and 12 are completely independent of one another except for Section 9 which uses some information from Section 8, but nevertheless can be read without knowing Section 8. So the reader has not to be overwhelmed by the length of this article, but pick what he is interested in. For Section 11, however, we recommend to read through Section 8 first. For the groups (1), (2) and in particular (3) we have included some of the proofs in the text although they are very often somewhere in the literature. The motivation is sometimes that the proofs themselves are very illuminating or just nice, that the proofs are hard to find or that the proofs presented here are, hopefully, easier or better to understand than the one in the literature. It is certainly possible to read only the definitions, lemmas and theorems and skip all the proofs or additional information. We recommend this in particular to the fifth group of readers, and also the last episode of Star Trek - The Next Generation, we believe the title is "All good things" (in German TV "gestern, heute, morgen"). Although the Hst of references is quite long, it may well have happened that we have not cited a paper which should appear there, and we apologize for that.

738

W. Luck

This survey contains also mini-surveys on the following topics: 3-manifolds in Section 3, amenable groups in Section 4, residually finite groups in Section 5, Kahler manifolds in Section 7 and analytic Ray-Singer torsion and Reidemeister torsion in Section 9. There are other survey articles on topics of this survey article or related topics, for instance [42], [109, Section 8], [144,158,171,203,226,234]. We mention that we have for simplicity restricted ourselves to the von Neumann algebra of a group. One can formulate a lot of the results also for arbitrary von Neumann algebras. Group means always discrete group. We have restricted ourselves to regular coverings, applications of L^-cohomology to other topics are very briefly mentioned in Section 12. We wish to thank the Max-Planck-Institut in Bonn for its hospitality while parts of this article were written, John Lott for a lot of discussions about this topic and Clemens Bratzler for reading through the manuscript. Miinster, Wolfgang Liick, February 1999 1. L^-Betti numbers for CW-complexes of finite type In this section we introduce L^-Betti numbers for regular coverings of CW-complexes of finite type and discuss their main properties. Let Z be a connected CW-complex of finite type. Finite type means that each skeleton of X is finite, but X may have infinite dimension. The pth Betti number bp{X) is defined by the rank of the finitely generated abehan group Hp(X) given by the cellular or, equivalently, singular homology. In algebraic topology it has turned out to be useful to improve classical invariants such as Euler characteristic and signature by passing to the universal covering and taking the action of the fundamental group into account. This leads, for instance, in the case of the Euler characteristic to the finiteness obstruction, and in the case of the signature to surgery obstructions such as symmetric signatures. We want to apply this strategy to Betti numbers. The following naive approach does not work. One could think of applying an appropriate notion of dimension for modules over the integral group ring ZTT of the fundamental group n to the homology Hp(X) of the universal covering. Notice that the TT-action on X induces a ZTT-module structure on Hp(X). The problem is that in general IJTX is not Noetherian. Hence Hp(X) is not necessarily finitely generated although X is of finite type and therefore the pth module Cp(X) of the cellular ZTT-chain complex is finitely generated over Z;r. So the dimension may be infinite. Moreover, it is not at all clear that the dimension is additive. The reason why a lot of algebraic manipulations work nicely for the complex group ring of a finite group is that this ring is semi-simple. It has this property because there are enough projections, this being a consequence of the fact that C F is a Hilbert space. Namely, an ideal / in C r is a direct summand because it has an orthogonal complement. This property does not hold for infinite groups. However, if one enlarges the complex group ring to the von Neumann algebra, all the convenient properties of the complex group ring of a finite group carry over to arbitrary groups. This motivates the following definitions. If r is a group, define l^iF) by the Hilbert space of square-summable formal sums ^y^r^yY with complex coefficients Xy. Square-sunmiable, of course, means

L^-invariants of regular coverings of compact manifolds and CW-complexes ^ y - | A y p < o o and the inner product is given by

KeF

yer

'

yer

The group von Neumann algebra of F is defined by the space of F-equivariant bounded operators from l^(r) to itself

A/'CD = B{lHnj\nf.

(1.1)

where /^(F) is equipped with the obvious left T-action. This is not the standard definition, but equivalent to it. The standard trace of the von Neumann algebra is defined by tr = tr^rcD ••^(^) ^ C,

f ^

{f(e), 4(2)(r)'

where e e F C l^(F) is the unit element. This trace extends to matrices tr:M{n,n,Af{F))

-^ C

(1.2)

by sending a matrix to the sum of the traces of the diagonal entries. A Hilbert Af(F)-module is a Hilbert space V together with a left action of F by linear isometrics such that there is a Hilbert space H and a F-equivariant isometric embedding of V into the tensor product of Hilbert spaces l^(F) 0 H. The isometric embedding is not part of the structure, only its existence is required. An example is l^iF) itself with the obvious left F-action. A Hilbert Af(F)-modu[Q is finitely generated if there is a surjective bounded F-equivariant operator from l^(F)^ = 0 " ^ i l^iF) onto V for an appropriate positive integer n. This is equivalent to the existence of an isometric F-equivariant embedding of V into fi{FY for an appropriate positive integer n and to the existence of an orthogonal F-equivariant projection ^x\l^{FY -> fi{FY whose image is isometrically jT-isomorphic to V for an appropriate positive integer n. A morphism of Hilbert J\f{F)-modules f :U ^^ V is a bounded F-equivariant operator. We call / a weak isomorphism if its kernel is trivial and its image is dense. The polar decomposition of a weak isomorphism / looks like / o | /1 where | / 1 : L^ ^- L^ is a positive morphism and /: L^ ^- V an isometric isomorphism. In particular U and V are isometrically isomorphic if there is a weak isomorphism from U to V. A sequence of Hilbert Ar(r)-modules is called exact if / is injective, im(/) = ker(p) and p is surjective. It is called weakly exact if / is injective, clos(im(/)) = ker(/7) and clos(im(/7)) = W. DEFINITION 1.3. Let V be a finitely generated Hilbert A/'(r)-module. Define its von Neumann dimension by dim(y) = dimAr(r)(V) = tr(pr) G [0, oo), where pr:l^(F)^ -^ f-{Fy is any orthogonal T-equivariant projection whose image is isometrically T-isomorphic to V.

739

740

W. Luck

It is not hard to check that the definition above is independent of the choice of projection. The elementary proof of the next result is left to the reader. It follows from the general properties of the universal center valued trace of a finite von Neumann algebra [131, Theorem 8.2.8, p. 517, Proposition 8.3.10, p. 525 and Theorem 8.4.3, p. 532]. 1.4. LetU,V and W be finitely generated Hilbert J\f(r)-modules. Then (1) Faithfulness: 6ivcij\f(^r){U) = 0 if and only ifU = 0; (2) Monotony: If U C V then

LEMMA

dim^rcnC^) < dim;v'(r)(^); (3) Continuity: If U\ D U2 D - -- is a nested sequence of Hilbert ofU, then

J\f(r)-submodules

dim;s/(r)( P i ^« 1 = ^™ dim;v'(r)(^«); \n=\

(4) Weak exactness: IfO->U^^ dimXiniV)

V ^ - W -> 0 is weakly exact, then

= dim^N/cnC^) + dimyv-cnC^)-

We will extend this notion to arbitrary modules over AfiF) in Section 10. DEFINITION 1.5. Let X be a (not necessarily connected) CW-complex of finite type. Let p:X -> X be 3. regular covering of X with group of deck transformations F acting from the left. Define the cellular L^-chain complex of X by the chain complex of finitely generated Hilbert J\f(F)-modules

cP(x)=/2(r)0zrC*(x). The cellular L? -cochain complex is defined by C(*2)(X)=homzr(c4x),/2(r)) where for y G F a n d / G C*2x(X) the element/ • / sends jc G C*(X) to /(x)}/~*. Define the L?-homology of X by Hf\x)=Ur{cf)/cXos(im{c%)), where cj, is the differential of Q yX). The definition of L?-cohomology is analogous. Define the pth L?-Betti number of X by bf{X)=dimMin{Hf\x))=dimMin{Hj;^^{X)),

L -invariants of regular coverings of compact manifolds and CW-complexes

741

The decisive difference between L^-homology and the T^-equivariant homology with coefficients in l^iF) viewed as ZF-module is that we divide by the closure of the image of the corresponding differential and not only by the image itself. The reason is that in the L^-setting we want to keep the Hilbert space structure coming from the cellular L^chain complex. In order to guarantee completeness we must divide by a closed subspace. We will investigate the difference between these two homologies later when we introduce Novikov-Shubin invariants in Section 8 which measure this difference. Hilbert complexes in general (not necessarily over a finite von Neumann algebra) are treated in [39]. If we deal with a smooth manifold M, then these definitions are understood for the CWcomplex structure given by some smooth triangulation. We will prove in Theorem 1.7 that the choice of triangulation does not matter because two triangulations of M give homotopy equivalent CW-complexes. In the sequel we will denote by X the universal covering with the fundamental group n\(X) as group of deck transformations F, provided that X is connected. Notice that both Hp (X) and HF2AX) are isometrically F-isomorphic to the kernel of the combinatorial Laplace operator A . = {cfrocf+c%o{c%r:Cf{X)^Cf{X).

(1.6)

The elementary proof can be found in [163, Theorem 3.7, p. 230]. Hence there is no difference between homology and cohomology in the L^-setting. Next we discuss the main properties of L^-Betti numbers, in particular in comparison with the ones of the ordinary Betti numbers. THEOREM

1.7.

(1) Homotopy invariance Let X and Y be regular coverings of CW-complexes X and Y of finite type with the same group F of deck transformations. Let f :X ^> Y be a F -equivariant map. If f is a homotopy equivalence, then

bf{X)^bf{Y)

forO^p.

If f is d-connected, i.e., f induces an isomorphism on 7tn for n < d and an epimorphism on TZd, then bf\x)

= bf\Y)

bf\x)

^

forp
bf{Y).

(2) Euler-Poincare formula Let X be a regular covering of a finite CW-complex X. Let X(X) = ^ ( - l ) ^ . ^ p ( X ) € Z

742

W. Luck be the Euler characteristic ofX where Pp{X) is the number of p-cells ofX. Then

xix) = Y^{-\y^bf[xy (3) Poincare duality Let M be a regular covering of the closed manifold M of dimension n. Then

bf{M) = b'^l^{M); (4) KUnneth formula Let X and Y be CW-complexes offinite type. Let X and Y be regular coverings ofX and Y. Then X x Y is a regular covering ofXxY and b(^\XxY)=

J]

bf\x).b<^\Y)

forn^O;

p-\-q=n

(5) Morse inequalities Let X be a regular covering of a CW-complex X of finite type. Let fi (X) be the number of p-cells in X. Then n

^ ( - i r - / ^ . bf[x) /?=0

n

^ Y.^-\r-P

. Pp{X) forn > 0;

p=0

(6) L ^ -Hodge-deRham decomposition Let M be a covering of the oriented closed Riemannian manifold M with deck transformation group F. Let 71^2)^^) ^^ ^^^ space of harmonic smooth L^-pforms on M, i.e., smooth p-forms co on M such that fj^co A ^co is finite and co lies in the kernel of the Laplace operator with respect to the induced Riemannian metric on M. Then integration defines an isomorphism of finitely generated Hilbert MiF)-modules <2)(^)^^(2)(^);

(7) Multiplicative property for finite coverings Let X be a CW-complex of finite type and p'.X^^Xbea regular covering with group of deck transformations F. Let FQ C F be a subgroup of F of finite index n. We obtain a regular covering denoted by X by X -^ X/FQ. Notice that the coverings X and X have the same total spaces but different groups of deck transformations. Then bf\x)=n.b^^\x)

forp^O;

L -invariants of regular coverings of compact manifolds and CW-complexes (8) L?-Betti numbers for finite groups F Let X be a CW-complex of finite type and let p'.X -^ X be a regular covering with group of deck transformations F of finite order \F\. Then:

bf{'^) = ~bp{x)

forp^O;

(9) Zero-th L^-Betti number Let X be a connected CW-complex of finite type and let p\X ^^ X be a regular covering with group of deck transformations F and connected X. Then 1 if\F\
b'.'\X): 0

otherwise',

(10) S^ -actions and L^-Betti numbers Let M be a connected closed manifold with S^ -action. Suppose that for one orbit S^ /H {and hence for all orbits) the inclusion into M induces a map on 7t\ with infinite image. (In particular the S^-action has no fixed points.) Then b^^\M)=0

forp^O.

PROOF. (1) Because of the Equivariant Cellular Approximation Theorem we can assume that / is cellular. Since X and Y are free /^-CW-complexes the map / is even SL Fhomotopy equivalence. Because of the Equivariant Cellular Approximation Theorem there is a cellular F-map g:Y -^ X such that there is a cellular F-homotopy between / o ^ (respectively g o f) and the identity. One easily checks that two cellular F-maps which are connected by a cellular F-homotopy induce the same map on L^-homology. Now the claim for a homotopy equivalence / follows. The more general case of a J-connected /^-map / is proven in [148, Lemma 3.3]. (2) This follows as in the classical situation, where x (^) is expressed in terms of (ordinary) Betti numbers, from the fact that the von Neumann dimension is weakly additive (see Theorem 1.4.4). (3) Follows from the Poincare ZF-chain homotopy equivalence [247, Theorem 2.1, p. 23]

p|[M]:C'^-*(M)->C*(M). (4) Follows from the isomorphism of cellular chain complexes C{x)^zC{Y)^C{XxY). See also [255, Corollary 2.36, p. 181]. (5) Analogous to (2). (6) Is proven by Dodziuk [72].

743

744

W Luck

(7) If y is a finitely generated A/'(r)-module and res(y) its restriction to N'iFo) which is a finitely generated HilbertA/'(/o)-module, then: dim7V'(ro)(res(V)) = n • dim7\^(r)(V). (8) Follows from (7). (9) If F has finite order, this follows from (8). If F is infinite, this follows from the fact t h a t / ^ ( r ) ^ is trivial. (10) is proven in [163, Theorem 3.20, p. 235]. D More information about Morse inequalities and L^-invariants can be found in [89,90, 157,172,173,196,197,234]. EXAMPLE 1.8. A good source of well understood examples is given by the special case where F is the free abelian group Z^ of rank r. On one hand everything becomes simple, on the other hand one can already see some of the important phenomenons in this special case. See also [57, Section 5], [76]. One simplification comes from Fourier transformation. Namely, we obtain a natural isometric Z^-equivariant isomorphism

where L^(T') is the Hilbert space of L^-functions on ^ ^ Let L'^iT'^) be the C*-algebra of essentially bounded measurable functions on T^. Then we obtain an isomorphism of C*-algebras

M: L ^ ( r ) ^ M{r) = B{L^{T'), L^{r)f^

f^Mf,

where Mf : L?{T^) -> L^(T^) sends g to the function / • g which assigns to z G T^ the element f{z) • g(z). A morphism of Hilbert A/'(Z'')-modules L?-{T^) -> L^(T^) is given by Mf for some / G L^(T^). It is a weak isomorphism if and only if /~^ (0) is a set of measure zero, and it is an isomorphism if for some ^ > 0 the set {z G T^ | | / ( z ) | < e] has measure zero. In particular we see concrete examples of weak isomorphisms which are not isomorphisms. An important example is given by M^.-i. The operator Mf is positive if and only if / takes values in the non-negative real numbers. In this case the spectral family {f";^ | A, G R} is given by Ex = M-^^ where xx is the characteristic function of the set {z G r^ I / ( z ) < A,}. The von Neumann trace tr;v'(Zo becomes

Let X ^- Z be a regular covering of a CW-complex of finite type with Z^ as group of deck transformations. Let Z[Z'^](0) be the quotient field of the integral group ring of Z^.

L^ -invariants of regular coverings of compact manifolds and CW-complexes

745

Let dimz[zn(0) i^pi^) ^Z[zn ^[^^^1(0)) be the dimension of the finite-dimensional vector space of the quotient field. Then we get [157, Example 4.3] bf{X)=dimz\zr],,,{Hp{X)

(8)z[zn ^[^'1(0)).

1.9. The L^-Hodge-deRham decomposition of Theorem 1.4.6 proves that for a regular covering M -^ M of a closed Riemannian manifold M the L^-Betti numbers have the following analytic interpretation. Let L^QP(M) be the Hilbert space of all square-integrable C-valued p-forms on M. This is the Hilbert space completion of the space C^QP(M) of smooth C-valued p-forms on M with compact support and the standard L^-pre-Hilbert structure. Since M is complete (with respect to the lifted Riemannian metric), the Laplace operator A^ is essentially selfadjoint in L?Q^{M), i.e., its closure with respect to the domain C^^P(M) is a self-adjoint operator on L^QP(M) [55]. Let Ap = f XdE^ be the spectral decomposition with right-continuous spectral family [E^ \ X eR}. Let E[Oc,y) be the Schwartz kernel of E[. Since E[(X,X) is an endomorphism of a finite-dimensional complex vector space, its trace trc is defined. Let T be a fundamental domain for the F-action on M, i.e., an open subset T in M such that {jy^r y clos(jr) = M and y j r n ^ = 0 for y G r with y 7^ 1 [210, Section 6.5]. Then the analytic pth spectral density function is defined by REMARK

FP(X) = j irc{E^(x,x))dx,

XeR.

(1.10)

We will later investigate this spectral density function more closely. We mention the equality b^^\'M) = FP(0).

(1.11)

By means of a Laplace transformation we obtain the equality b^^UJi) = lim f tTcU-'^p(x,x))dx,

(1.12)

where Q~^^P (X, J ) denotes the heat kernel on M, i.e., the Schwartz kernel of QT^^P . All the definitions and results above extend to pairs of proper cocompact F-CWcomplexes and manifolds with boundary and proper cocompact T-actions [163, Sections 3 and 5]. The L^-Hodge-deRham decomposition for manifolds with boundary is proven in [228].

2. Basic conjectures In this section we state the (in our view) most important conjectures concerning L^-Betti numbers. They motivate a lot of the work done on this subject. We also give a list of

746

W. Luck

theorems which give evidence for them. Results of Sections 3, 4 and 7 also prove these conjectures in special cases. More questions and conjectures will be discussed in other sections. CONJECTURE 2.1 {Rationality ofL^-Betti numbers). Let T be a group and A be a (m, n)matrix with coefficients in ZF. Let f-.fir)"^ -> f{rY be the F-equivariant bounded operator induced by right multiplication with A. Then (1) dim7V'(r)(ker(/)) is rational; (2) Let /: be a positive integer such that the order of any finite subgroup of F divides k. Then k • dim7\^(/-)(ker(/)) is an integer; (3) If F is torsionfree, then dimj\f(^f^(kcY(f)) is an integer.

Conjecture 2.1 above has the following equivalent reformulations provided that F is finitely presented. LEMMA 2.2. The following statements are equivalent for a finitely presented group F: (1) F satisfies Conjecture 2.1.1 {respectively 2.1.2, respectively 2.1.3); (2) If X is a connected CW-complex of finite type with F as fundamental group, (2) —

then bp {X) satisfies the corresponding statement of Conjecture 2.1.1 {respectively 2.1.2, respectively 2.1.3) for all p^O; (2) "^

(3) If M is a closed manifold with F as fundamental group, then bp (M) satisfies the corresponding statement of Conjecture 2.1.1 {respectively 2.1.2, respectively 2.1.3) for all p^O. PROOF. Obviously (1) implies (2) and (2) implies (3) so that it remains to prove that (3) implies (1). The key observation is the following. Let A be a (m, /i)-matrix over ZF and J ^ 3 be an integer. Choose a finite connected 2-dimensional CW-complex X with F as fundamental group. By attaching n {d — l)-cells to X with a trivial attaching map and m dcells one can construct a finite CW-complex Y such that the Jth differential of the cellular Zr-chain complex C{Y) is, for J ^ 4, given by A and for J = 3 by the map induced by A composed with the canonical inclusion/^(r)'^ -> fi{FY^f'{FY where nMs the number of 2-cells in X. Let / : 1^{F)^ -> f-{FY be the bounded T-equivariant operator induced (2) —

by A. Since Y has no {d + 1)-cells, the kernel of / is just H^ (7). Next one can embed the finite J-dimensional CW-complex Y into R^^+^. Let M be the boundary of a regular neighborhood of X [221, Chapter 3]. Then there is a (6? -f-1)-connected map M ^^ X and because of Theorem 1.7.1

b? (^) = b?{^)= dim(ker(/)).

D

REMARK 2.3. If Conjecture 2.1 holds for all finitely generated groups it holds for all groups. Namely, given a (m, n) matrix A with coefficients in Z r , it suffices to look at the subgroup F' generated by those elements which appear in one of the entries of A with non-zero coefficients. Namely, A can be viewed as a matrix over ZF' and the dimension over F of the associated F-equivariant bounded operator agrees with the one over F'.

L -invariants of regular coverings of compact manifolds and CW-complexes

747

Lemma 2.2 remains true if we substitute the assumption finitely presented for F by finitely generated and substitute the universal coverings X (respectively M) by regular coverings X ^^ X (respectively M -^ M) for a connected finite CW-complex (respectively closed manifold M). The modification in the proof is as follows. Choose a finitely generated free group F together with an epimorphism p:F ^^ F. Lift A to a matrix A^ over Z[F] and construct X, Y and M as explained in the proof of Lemma 2.2 for F and A^ Then take the coverings of Y (respectively M) associated to p with F as group of deck transformations. Hence Conjecture 2.1 is true for all groups F and matrices A if and only if for all regular coverings M ^^ M oi closed manifolds M with a finitely generated group of deck transformations by(M) satisfies the corresponding statement. The question of whether the third statement in Lemma 2.2 is true was asked by Atiyah [3, p. 72]. Next we show that Conjecture 2.1 implies the Zero-Divisor-Conjecture. For a discussion of the Zero-Divisor-Conjecture and related conjectures we refer to [138, p. 95]. LEMMA 2.4. If the group F satisfies Conjecture 2.1.3, then it also satisfies the ZeroDivisor-Conjecture which says: The rational group ring QF has no non-trivial zerodivisors if and only if F is torsionfree. PROOF. Suppose that F is not torsionfree. Let ^ G T be an element of finite order |g| 7«^ 1. -1^1 ^^ ^^ ^ zero-divisor in QF since it satisfies Then the norm element N = Ylk=\

A^-(|^|-A^) = 0.

Suppose that F is torsionfree. Let x G Q F be a zero-divisor. We have to show, under the assumption that Conjecture 2.1.3 is true, that x is trivial. By multiplying x with an appropriate integer, we can assume without loss of generality that x belongs to Z F . Right multiplication with X induces a F-equivariant bounded operator r^ : l^(F) -^ l^(F). Conjecture 2.1 implies that dimj\fi^r) (ker(rjc)) is an integer. As x is a zero-divisor, the kernel of rx is not trivial and hence dim7v'(r)(ker(rjc)) is not zero. Since dim;v'(r)(ker(rx)) is bounded by the dimension of/^(F), which is 1, we get dim;\^(/-)(ker(rjt)) = 1 = dimj^^f){l^(F)). As the kernel of r^ is a closed subspace of l^(F), this implies that rx is the trivial map. Hence x is zero. This shows that the rational group ring has no non-trivial zero-divisors. D Let C be the smallest class of groups which (i) contains all free groups, (ii) is closed under directed unions and (iii) satisfies G eC whenever G contains a normal subgroup H such that H belongs to C and G/H is elementary amenable. We recall that the class of elementary amenable groups is defined as the smallest class of groups which contains all finite and all abeUan groups, and is closed under taking subgroups, forming factor groups, group extensions, and upwards directed unions. THEOREM

2.5 (Linnell [140]). Conjecture 2.1 holds for the class C of groups.

748

W. Luck

The key result in [140] is that for a torsionfree group F in the class C there is a division ring D{r) satisfying CF c D{r) C U(r), where U{r) is the algebra of closed densely defined operators /^(F) -> /^(F) which are affiliated to the group von Neumann algebra. Linnell uses the Fredholm technique developed by Connes for his proof that the reduced C*-algebra of a free group has no non-trivial projections [62, Section 7], [83], Moody's induction theorem [178, Theorem 1] and Cohn's theory of universal fields of fractions [61], This indicates that there must be a connection between Conjecture 2.1 and the Baum-Connes Conjecture for the topological ^-theory of the reduced C*-algebra of a group [13, Conjecture 3.15, p. 254] and the Isomorphisms Conjecture for the algebraic A'-theory of the integral group ring of Farrell and Jones [93, 1.6, p. 257]. See [68] for a unified treatment of the last two conjectures and see [213] for more details on this connection and a general strategy based on Linnell's work how to approach Conjecture 2.1. The Baum-Connes Conjecture says that one can compute the topological A^-theory of the reduced C*-algebra of a group F by a complicated induction process from the topological A'-theory of the complex group rings of all finite subgroups of F. Conjecture 2.1 can be interpreted similarly, namely all the possible dimensions of kernels of F-equivariant bounded operators f-{F)^ —^ l^(Fy which are induced by matrices over ZF are coming from the dimensions of kernels of T-equivariant bounded operators 1^{F)^ -^ f-^FY which are induced by matrices over ZG for all finite subgroups G C F. Notice that the dimension of such operators coming from a finite group G are rational numbers which become integral when multipHed with the order of G. The missing link seems to be the not at all understood passage from finitely presented ZF-modules to modules over the C*algebra of F. The connection between the generalization of the Euler Poincare formula for finitely dominated CW-complexes and the Bass Conjecture [12, p. 156] which is related to the Baum-Connes Conjecture and the Isomorphisms Conjecture for the algebraic ^-theory is treated by Eckmann [82]. CONJECTURE 2.6 (Singer Conjecture). If M is a closed aspherical Riemannian manifold of dimension n, then

bf\M)

=0

forlp^n.

If additionally n = 2d, then (-l/.X(M)>0. The conjecture above was originally only formulated for closed Riemannian manifolds with non-positive sectional curvature. Notice that any such manifold is aspherical by Hadamard's Theorem [99, 3.87, p. 134]. We recall that a space X is aspherical if its universal covering is contractible, or equivalently, all the higher homotopy groups of X are trivial. We will see that the Singer Conjecture 2.6 is true if n\ (M) contains a normal nontrivial amenable subgroup in Section 4, and if M has dimension 3 and is not exceptional in Section 3. The Singer Conjecture 2.6 is true if M is a symmetric space of non-compact type. It is also true if M carries a non-trivial 5^-action, because then the inclusion of an orbit into M induces a map on the fundamental groups with infinite image [66, Lemma 5.1, p. 242 and Corollary 5.3, p. 243] and Theorem 1.7.10 applies.

L^ -invariants of regular coverings of compact manifolds and CW-complexes

749

CONJECTURE 2.7 (Hopf Conjecture). If M is a closed 2^-dimensional Riemannian manifold of negative sectional curvature, then

b^J\M) >0;

(-l/.x(M)>0.

In the Hopf Conjecture 2.7 above, the part about the Euler characteristic goes back to Hopf. The statements about the Euler characteristic in Conjecture 2.6 and 2.7 follow from the one about the L^-Betti numbers by the Euler-Poincare formula x (^) = ^p>o(~^)^ * of Theorem 1.7.2. Conjecture 2.7 has been proven by Dodziuk for a hyperbolic closed Riemannian manifold. Namely, we have already mentioned in Theorem 1.7 that there is L^-Hodge-deRham decomposition [72]. Since the universal covering of a closed /i-dimensional hyperbolic manifold is isometrically isomorphic to the n-dimensional hyperbolic space H", and the von Neumann dimension of a finitely generated Hilbert ATcr)-module is zero if and only if the module is zero, the pth L^-Betti number of M is trivial if and only if the space of harmonic L^-integrable /?-forms on M.^ is trivial. This space is computed in [73] using the rotational symmetry of M.^. Donnelly and Xavier [79] have proven for a closed ndimensional Riemannian manifold M whose sectional curvature is pinched between — 1 and Dn for some real number — 1 ^ D„ < — ^^~ A that Z?^^\M)

In particular, the Singer Conjecture 2.6 is true for such a manifold if n is even. This result has been improved by Jost and Yuanlong [129]. They assume that the sectional curvature of M satisfies —a^^K^O and the Ricci curvature is bounded from above by —b^ for positive constants a, b and show that

bf\M)

=0

for /7 / | , 2pa ^ b.

We will see in Section 7 that the Hopf Conjecture 2.7 is true if M is a Kahler manifold. Without some finiteness conditions on X, such as being the total space of a regular covering over a finite CW-complex, Conjecture 2.6 becomes false. This follows from the work of Anderson [2] where the non-vanishing of the (reduced) L^-cohomology of perturbations of the hyperbolic metric on the hyperbolic space is proved.

3. Low-dimensional manifolds In this section we give information about the L^-Betti numbers of universal coverings of compact manifolds of dimension < 3. We will only consider orientable manifolds since one gets the general case by passing to the orientation covering and the multiplicative property of the L^-Betti numbers of Theorem 1.7.7.

750

W. Luck

EXAMPLE 3.1. We begin with a compact connected 1 -dimensional manifold M.\fM has boundary, it is [0, 1] and hence homotopy equivalent to a point, and its L^-Betti numbers agree with the (ordinary) Betti numbers of the one-point-space, i.e., they are trivial except the 0th one which is 1. If M has no boundary, then M is 5 ^ Since there is a covering S^ -> S^ of degree d for d ^ 2, we conclude from the multiplicative property in Theorem 1.7.7

^(^2)(^i)^0

forp^O.

It is illuminating to compute bp (S^) directly. One easily checks that the cellular L^9

t-\

chain complex is concentrated in dimension 0 and 1 and its first differential / (Z) —> /^(Z) is given by multiplication with the element r — 1 for / a generator of the fundamental group 7t\(S^) =Z. Its Fourier transformation is the bounded Z-equivariant operator M,_i: L\S')

^

L\S'),

f(z)

H^ (z - 1) • / ( z ) ,

where we regard 5^ as a subset of the complex numbers C. Obviously the kernel of M^-i and hence //[ ( 5 ^ is trivial. One easily checks that H^ (S^)is the kernel of M* |, which is the kernel of M^-i _i and hence trivial. 3.2. Let F^ be the orientable closed surface of genus g with d embedded open 2-disks removed. From Theorem 1.7 and the fact that a compact surface with boundary is homotopy equivalent to a bouquet of circles, one derives EXAMPLE

^ ( 2 ) ( ^ \ _ f 1 ^ = 0 , J = 0,1, ^ ^ s/ 10 otherwise, ^(2)/^.^

1 ^ s)

fO

^ = 0 , ^ = 0,1,

| j + 2(g-l)

otherwise,

^(2)/^\ = n 2 ^ ^^

lo

g = 0,J = 0, Otherwise.

In the sequel 3-manifold means a compact connected orientable 3-dimensional manifold. Notice that in dimension 3 there is no difference between topological, PL- or smooth manifolds [176,191]. We want to compute the L^-Betti numbers of the universal covering of a 3-manifold. For this purpose we have to recall some basic facts about such manifolds which are interesting in their own right. For more information about 3-manifolds we refer to [116,231,240]. A 3-manifold M is prime if, given a decomposition M\ ft M2 of M as a connected sum, M\ or M2 is homeomorphic to S^. It is irreducible if every embedded bicollared 2-sphere bounds an embedded 3-disk. Any prime 3-manifold is irreducible or is homeomorphic to S^ X S^ [116, Lemma 3.13]. One can write a 3-manifold M as a connected sum M = MittM2tt---ttM„

L -invariants of regular coverings of compact manifolds and CW-complexes

751

where each Mj is prime, and this prime decomposition is unique up to renumbering and oriented homeomorphism [116, Theorems 3.15, 3.21]. By the Sphere Theorem [116, Theorem 4.3], an irreducible 3-manifold is aspherical if and only if it is a 3-disk or has infinite fundamental group. A properly-embedded orientable connected surface in a 3-manifold is incompressible if it is not a 2-sphere and the inclusion induces an injection on the fundamental groups. One says that 9M is incompressible in M if and only if 9 M is empty or any component C of 9M is incompressible in the sense above. An irreducible 3-manifold is Haken if it contains an embedded orientable incompressible surface. If M is irreducible, and in addition H\{M) is infinite, which is implied if 9M contains a surface other than 5^, then M is Haken [116, Lemma 6.6 and 6.7]. (With our definitions, any properly embedded 2-disk is incompressible, and the 3-disk is Haken.) We call a manifold hyperbolic if its interior admits a complete Riemannian metric whose sectional curvature is constant — 1. Provided that M has no boundary, this is equivalent to the statement that the universal covering with the lifted Riemannian metric is isometrically isomorphic to the hyperbolic space of the same dimension as M. We use the definition of a Seifert 3-manifold of [231, p. 429]. If Jt\ (M) is infinite, M is Seifert if and only if some finite covering of M is the total space of an 5^-principal bundle over a compact 2dimensional manifold. The work of Casson and Gabai shows that an irreducible 3-manifold with infinite fundamental group TT is Seifert if and only if n contains a normal infinite cyclic subgroup [98, Corollary 2, p. 395]. Next we mention what is known about Thurston's Geometrization Conjecture for irreducible 3-manifolds with infinite fundamental groups. Johannson [127] and Jaco and Shalen [125] have shown that, given an irreducible 3-manifold M with incompressible boundary, there is a finite family of disjoint, pairwise-nonisotopic incompressible tori in M which are not isotopic to boundary components and which split M into pieces that are Seifert manifolds or are geometrically atoroidal, meaning that they admit no embedded incompressible torus (except possibly parallel to the boundary). A minimal family of such tori is unique up to isotopy, and we will say that it gives a toral splitting of M. We will say that the toral splitting is geometric if the geometrically atoroidal pieces which do not admit a Seifert structure are hyperbolic. Thurston's Geometrization Conjecture for irreducible 3-manifolds with infinite fundamental groups states that such manifolds have geometric toral splittings. For completeness we mention that Thurston's Geometrization Conjecture says, for a closed 3-manifold with finite fundamental group, that its universal covering is homeomorphic to 5^, the fundamental group of M is a subgroup of S0(4) and the action of it on the universal covering is conjugated by a homeomorphism to the restriction of the obvious 50(4)-action on S^. This impHes, in particular, the Poincare Conjecture that any homotopy 3-sphere is homeomorphic to S^. Suppose that M is Haken. The pieces in its toral splitting are certainly Haken. Let A^ be a geometrically atoroidal piece. The Torus Theorem says that A^ is a special Seifert manifold or is homotopically atoroidal, i.e., any subgroup of n\ (N) which is isomorphic to Z x Z is conjugate to the fundamental group of a boundary component. Thurston has shown that a homotopically atoroidal Haken manifold is a twisted /-bundle over the Klein bottle (which is Seifert) or is hyperbolic.

752

W. Luck

Thus the case in which Thurston's Geometrization Conjecture for an irreducible 3manifold M with infinite fundamental group is still open is when M is a closed non-Haken irreducible 3-manifold with infinite fundamental group which is not Seifert. The conjecture states that such a manifold is hyperbolic. We want to compute the L^-Betti numbers of the universal covering of a 3-manifold under the assumption that Thurston's Geometrization Conjecture holds for the pieces in the prime decomposition with infinite fundamental group. First we deal with the case where the fundamental group of M is finite. In this case the L^-Betti numbers are the ordinary Betti numbers of M divided by the order |7r| of the fundamental group TT of M. If M is closed, M is homotopy equivalent to S^, and hence b^ (M) = b^ (M) = \7t\~^ and (2) —

bp (M) =OfoTp^O, 3. Suppose that dM is non-trivial. Then M is a connected sum of homotopy spheres and k disks for some positive integer k, and hence b^ (M) = \7T\~\ b^^\M) = (/: - 1) • ITT r U n d b^p\M) = 0 for /7 / 0, 2. Hence we only have to treat the case where 7T\ (M) is infinite. Let us say that a prime 3-manifold is exceptional if it is closed and no finite covering of it is homotopy equivalent to a Haken, Seifert or hyperbolic 3-manifold. No exceptional prime 3-manifolds are known, and Thurston's Geometrization Conjecture and Waldhausen's Conjecture that any 3-manifold is finitely covered by a Haken manifold imply that there are none. Notice that any exceptional manifold has infinite fundamental group. 3.3 (Lott and Liick [148, Theorem 0.1]). Let M be the connected sum Ml tt • • • tJ^r of (compact connected orientable) non-exceptional prime 3-manifolds Mj. Assume that 7t\ (M) is infinite. Then the L^-Betti numbers of the universal covering M are given by THEOREM

b'i\M)

= 0;

^ 1 fof>(M) = ( r - l ) - ^ ^ - ^ ^ - X ( M ) + | { C 6 ; r o O M ) | C ^ 5 2 } | ; Z,f >(M) = (r - 1) - X : 1 ^ ^ ^ + |{C € ;roOM) I C = 52}|; ^ f >(M) = 0. PROOF.

We give a sketch of the strategy of proof. Since the fundamental group is infinite, (2) "^

(7.) '^

we get Z?Q (M) = 0 from Theorem 1.7.9. If M is closed, we get b\^ (M) = 0 because of Poincare duality 1.7.3. If M has boundary, it is homotopy equivalent to a 2-dimensional CW-complex and hence b\^ (M) = 0. It remains to compute the second L^-Betti number, because the first one is then determined by the Euler-Poincare formula 1.7.2. It is not hard to find a general formula for the L^-Betti numbers of a connected sum in terms of the summands. With this formula we reduce the claim to prime 3-manifolds. Since a prime 3-manifold is either irreducible or ^* x S^, the claim is reduced to the irreducible case. If the boundary is compressible, we use the Loop Theorem [116, Theorem 4.2, p. 39] to reduce the claim to the incompressible case. By doubling M we can reduce the claim

L^-invariants of regular coverings of compact manifolds and CW-complexes

753

further to the case of an irreducible 3-manifold with infinite fundamental group and incompressible torus boundary. By the toral splitting and the assumptions about Thurston's Geometrization Conjecture we can reduce further to the claim that the L^-Betti numbers vanish if M is Seifert with infinite fundamental group or is hyperbolic with incompressible torus boundary. All these steps use the weakly exact Mayer-Vietoris sequence for L^(co)-homology of Cheeger and Gromov [52, Theorem 2.1, p. 10]. In the Seifert case we can assume by the multiplicative property (see Theorem 1.7.7) that M is an 5"'-principal bundle over a 2-dimensional manifold. Then we use induction over the cells of the base space and the fact that the L^-Betti numbers of S^ vanish. The hyperbolic case is reduced to the known closed case by a careful analysis at the boundary using the fact that such a hyperbolic manifold with incompressible torus boundary has finite volume. D Let Xvirt(7ri (M)) be the rational-valued group Euler characteristic of the group 7t\ (M) in the sense of [35, IX.7], [246]. Then the conclusion in Theorem 3.3 is equivalent to bf\M)

= -Xvirt(7ri(M));

bf\M)

=x(M)-Xvirt(7ri(M)).

This is proven in [148, pp. 53-54]. Notice that Theorem 3.3 proves Conjecture 2.6 and the third of the three equivalent assertions in Lemma 2.2 for compact 3-manifolds, provided that Thurston's Geometrization Conjecture or Waldhausen's Conjecture is true. Notice that this does not imply Conjecture 2.1 for r the fundamental group of a compact 3-manifold satisfying Thurston's or Waldhausen's Conjecture. 4. Aspherical manifolds and amenability This section is devoted to a result of Cheeger and Gromov about the vanishing of the L^Betti numbers of the universal covering of an aspherical CW-complex whose fundamental group contains a non-trivial normal amenable subgroup. Let / ^ ( F , R) be the space of bounded functions from F to M with the supremum norm. Denote by 1 the constant function with value 1. A group F is called amenable if there is a r-invariant linear operator /x: / ^ ( F , M) ^- R with /x(l) = 1 which satisfies inf{/(y) I K G r } < / / ( / ) ^ sup{/(y) \yer}

fovfe

l^iF),

The last condition is equivalent to the condition that /x is bounded and /x(/) ^ 0 if / ( y ) ^ 0 for ally G T. We give an overview of some basic properties of this notion. The class of amenable groups satisfies the conditions appearing in the definition of elementary amenable groups in Section 2, namely, it contains all finite and all abelian groups, and is closed under taking subgroups, forming factor groups, group extensions, and upwards directed unions [17, Proposition F.6.11, p. 309]. Hence any elementary amenable group is amenable. Recently Grigorchuk has constructed a finitely presented group which is amenable but not elementary amenable. Any group containing the free group on two letters Z * Z as subgroup is

754

W Luck

not amenable [17, Proposition F.6.12, p. 310]. There are finitely generated, but not finitely presented groups, which are not amenable but do not contain Z * Z [200]. However, no non-amenable finitely presented group is known which does not contain Z * Z. A useful geometric characterization of amenable groups is given by the Folner criterion [17, Theorem F.6.8, p. 308] which says that a finitely presented group F is amenable if and only if for any positive integer n, any connected closed Riemannian manifold M with fundamental group 7T\ (M) = F, and ^ > 0, there is a domain i? c M with (n — 1)-measurable boundary such that the (n — 1)-measure of 9 ^ does not exceed e times the measure of Q. Such a domain can be constructed by an appropriate finite union of translations of a fundamental domain if F is amenable. The fundamental group of a closed connected manifold is not amenable if M admits a Riemannian metric of non-positive curvature which is not zero everywhere [9]. A group is amenable if and only if all its finitely generated subgroups are amenable [17, Proposition F.6.11, p. 309]. Any finitely generated group which is not amenable has exponential growth [17, Proposition F.6.24, p. 318]. A group F is amenable if and only if the canonical map from the full C*-algebra of F to the reduced C*-algebra of F is an isomorphism [207, Theorem 7.3.9, p. 243]. A group F is amenable if and only if the reduced C*-algebra of F is nuclear [139]. For more information about amenable groups we refer to [206]. The next result gives a positive answer to the Singer Conjecture 2.6 for special fundamental groups. THEOREM 4.1 (Cheeger and Gromov [54]). IfX is an aspherical connected CW-complex of finite type such that its fundamental group contains a non-trivial normal amenable subgroup, then we get for the universal covering X

b^^\x)=0

forp^O.

In this section we will explain only one of the decisive steps in the proof of Theorem 4.1 in order to illustrate the meaning of the condition about amenability. We will complete the proof in Section 10. Let X be a finite CW-complex with regular covering X ^^ X with group of deck transformations F. Let C*(Z) be the dual cochain complex with complex coefficients hom^(C* (X), C) of the cellular chain complex C* ( X ) . An element in C^ (X) is a function u:Sp(X) -^ C from the set of /7-cells of X to the complex numbers. We call u square-summable if Xl^e5 (X) l"(^)l^ i^ finite. The square-summable elements form a subcomplex C*2)(X) c C*(X). We equip the chain modules of C*2)(X) with the obvious Hilbert space structure. This definition agrees with Definition 1.5. Let H'^2)(^ ) ^^ the Hilbert submodule of C.^x(X) consisting of those elements u which satisfy c^2)(^) = ^ and (C(^27 )*(")= ^-There are obvious maps '•" : < 2 ) ( ^ ) - Hl'2^iCU^)) = ^r2)_(^); jP:n^2){^) -^ HP{C*{X)) = HP{X;C). The first map (4.2) is always an isomorphism by an elementary argument.

(4-2) (4.3)

L -invariants of regular coverings of compact manifolds and CW-complexes

755

LEMMA 4.4 (Cheeger and Gromov [54, Lemma 3.1, p. 203]). The map jP of (4.3) is injective, provided that F is amenable. Let D C X be a fundamental domain for the F-action on X. Such a D is constructed by choosing for each closed p-ccW e e Sp(X) one Hft e e Sp(X) and taking the union of these lifts for all /? > 0. Because F is amenable, one can find a sequence of subcomplexes Xj c X and natural numbers rij and drij such that Xj is the union of rij translates with distinct elements in y^ e F for k = 1,2,..., rij of D, drij is the numbers of cells of D which meet the boundary (in the sense of point set topology) of Xj and PROOF.

drij

lim —^ = 0.

j-^oo

(4.5)

rij

Let KP C n^2)^'X) be the kernel of jP. We have to show that K^ is trivial. Let pr^^ (respectively p r p be the orthogonal projection C^2) ( ^ ) ~^ ^(2) ( ^ ) ^^^^ ^^ (respectively onto the complex subspace of square-summable cochains u which are supported on Xj, i.e., u(e) = 0 unless e belongs to Xj ). Let xi denote the characteristic function Sp(X) ^^ C for a given ^G Sp(X). We get dim7v-(r)(^^) = tr^(r)(pr^/0=

J2

{P^Kp (Xe), Xe)

eeSpiX)

=

J2 {P^Kp(Xe),Xe)= — ' J2 ^J'{P^Kp(Xl)^Xe) eeSp(D)

^

1

eeSp(D)

""' eSp{D)k=\

— ' ^

y2

i^^KP (Xe),Xe)= — ' trc(pry o pr^;,).

eeSp(Xj)

Since pr^* and pr^p have norm 1, we conclude 1 d\mjsfi^r){KP) ^ — • dimc(im(pr^ opr^^;,)).

(4.6)

We fix the orthogonal decomposition u = Ui -\-Ud^Ue e C.^x(x) c m a p ( 5 p ( X ) , C ) , where w/ is only supported on (closed) cells in the interior of XJ^UQ is supported on cells which meet the boundary of Xj and Ue is supported on cells which do not meet Xj (and hence do not meet the boundary of Xj ).

756

W. Luck

Now consider an element u e K^ such that Mg = 0. Choose i; G C^~'(X) such that ^p-x. cP'^H) -^ CP(X) maps i; to M regarded as an element in C ^ ( Z ) . Notice that prJ extends to a map pr^ : CP(X) -^ C^(X), namely pr^Cy) for y G C ^ ( Z ) sends a cell e G 5^^ ( X ) to y (^) if e belongs to Xj and to zero otherwise. From this description we conclude pr^(M) = m. The cochain pr^oc^^^Cf) vanishes on cells which do not meet Xj. The difference CP~^ O pr^~ {v) — pr^ o CP~^{V) is supported on cells which meet the boundary of Xj because the boundary of a cell which does not belong to Xj cannot lie in the interior of Xj. Hence we get 0 ^ (pr^^(w),pr;(w)) = (prJoc^-l(i;),w,)

= (prj-^u), (c^-i)*(«)) = (prj-^t;), 0) = 0. This implies that pry vanishes on K^ Pi {w G C/^.( X) | MQ = 0}. Since d i m c ( ^ ^ / ( ^ ^ n[ue

Cf2)(x) I wa = 0}))

^ dimc(Cf2)(x)/{w G Cf2)(x) I Ma =0}) = dnj, we conclude dimc(im(pr^opr^p)) ^ duj.

(4.7)

We get from (4.6) and (4.7) dimAA(r)M^—. Since the limit of the right hand side is zero by (4.5), we get dimA/-(r)(/r^)=0. This finishes the proof of Lemma 4.4.

D

Notice that Lemma 4.4 proves Theorem 4.1 in the special case where the fundamental group of X is itself amenable by the following argument. In order to show that bp {X) vanishes, we can pass to the (t/ + l)-skeleton F c X and prove that Z?}, (7) vanishes. Since X is aspherical, we conclude //^(F;C) = //^(X;C)=0.

L^-invariants of regular coverings of compact manifolds and CW-complexes

757

Now Lemma 4.4 implies Hl',^{Y)^0;

fef2,(F)=0.

If r contains a normal amenable infinite group A such that B A is of finite type, then Theorem 4.1 follows by the L^-version of the Leray-Serre spectral sequence (see, for instance, [239]) applied to the fibration BA -> BF -^ BF/A. We will give the proof in the general case, where no assumptions about BA are made, in Section 10. 5. Approximating L^-Betti numbers by ordinary Betti numbers In this section we get the L^-Betti numbers of a regular covering X -^ X of a CW-complex of finite type as the limit of the normalized ordinary Betti numbers of a tower of finite coverings X^ ^^ X which converges in some sense to X. Let X -> X be a regular covering of a CW-complex of finite type with group of deck transformations F. Suppose that F is residually finite, i.e., for each element y e F with y / 1 there is a homomorphism 0 : F ^- G to a finite group with 0(y) ^ 1. We will assume that F is countable. Under this assumption F is residually finite if and only if there is a nested sequence of normal in F subgroups • • • C Tm+i C F^ C • • - C Fo = F such that the index [F : Fm] is finite for all m ^ 0 and the intersection nm>0 ^^ ^^ ^^^ trivial group. We give some information about residually finite groups at the end of this section. Consider any such sequence (r^)m^o- Let pm'.Xm = Fm\X -^ X be the covering of X associated with Fm C F. Notice that this is a finite regular covering of X, and hence Xm is again of finite type. Denote by bp{Xm) the (ordinary) pth Betti number of X^. THEOREM

5.1 (Liick [156]). Under the conditions above we get for all p^O

The inequality l i m s u p ^ ^ ^ TrTT^ ^ ^p (X) for X a closed manifold is discussed by Gromov [109, 0.5.F, 8.A] and is essentially due to Kazhdan [132]. The paper of Donnelly [77] deals with the operator /(Z\o) acting on 0-forms for a function / G C ^ ( R + ) . Theorem 5.1 is proven by Yeung [254] in the special case of a closed Kahler manifold with negative sectional curvature and by DeGeorge-Wallach [102,103] in the special case of a closed locally symmetric space of non-compact type. In the last case all the L^-Betti numbers vanish, so one gets lim —

= 0.

m ^ o o [F : Fm]

This leads to the question of how fast this sequence goes to zero; in other words, one wants to know the largest e for which ,. bpjXm) „ lim — ;— = 0

m^^[F'.Fm]^-' is true. Such questions are treated in [227,251,252].

758

W. Luck

The general case of a CW-complex X of finite type is proven in [156]. There actually the entire spectral density function of the combinatorial Laplace operator on the cellular L?chain complex of X is approximated by the spectral density function for the combinatorial Laplace operator on the cellular chain complexes of the various Xm • The spectral density function for Xm simply encodes the eigenvalues and their multiplicities of the Laplace operator because Xm is compact, whereas the one for X is more compHcated as, in general, X is not compact and the spectrum is not discrete. In some sense we try to approximate continuous information by discrete data. The philosophy is that the tower of finite coverings Xm converges to X. The inequahty Hmsup^^^ T r T ^ ^ Z?}, (X) is the easier part of the proof of Theorem 5.1. The main trick in the proof of the other inequality is not to forget and to use essentially the fact that the combinatorial Laplace operator on the cellular chain complex already lives over the integral group ring. This allows the use of the obvious inequality \n\ ^ 1 for an integer n different from zero. The results for the combinatorial Laplace operator on the cellular chain complexes carry over to the analytic Laplace operator acting on smooth p-forms on X provided that X is a compact smooth manifold. One can formulate Theorem 5.1 for any elliptic differential operator on a closed smooth Riemannian manifold X if one uses on each Xm and on X the lifted Riemannian metric and elliptic differential operator and substitutes the Betti numbers by the dimensions of the kernels. It seems not hard to prove the analogue of the inequality lim sup^^.^^ \'r-r'\ ^ ^P (^)» but it is not known whether the equality holds. The question is whether the result is only true for the Laplacian because it has a cellular analogue which allows us to use the fact that everything already lives over the integral group ring. Again we see from this discussion as we have mentioned in Section 2 that the passage from the integral group ring of F to the reduced C*-algebra, or even the von Neumann algebra of F, plays a fundamental role and any understanding of it seems to give new results. Finally we collect some basic facts about residually finite groups in order to explain how restrictive the assumption is that F is residually finite. For more information we refer to the survey article of Magnus [169]. The free product of two residually finite groups is again residually finite [60, p. 27], [113]. A finitely generated residually finite group has a solvable word problem [183]. The automorphism group of a finitely generated residually finite group is residually finite [14]. A finitely generated residually finite group is Hopfian, i.e., any surjective endomorphism is an automorphism [168], [195, Corollary 41.44]. Let T be a finitely generated group possessing a faithful representation into GL{n, F) for F a field. Then F is residually finite [168], [248, Theorem 4.2]. The fundamental group of a compact 3-manifold whose prime decomposition consists of manifolds which have finite fundamental groups, or are non-exceptional in the sense of Section 3 (i.e., which are finitely covered by a manifold which is homotopy equivalent to a Haken, Seifert or hyperbolic manifold), is residually finite [117, p. 380]. Let T be a finitely generated group. Let F^^ be the quotient of F by the normal subgroup which is the intersection of all normal subgroups of F of finite index. The group F^^ is residually finite and any finite-dimensional representation of F over a field factorizes over the canonical projection F -^ F^f. The upshot of this discussion is the slogan that the fundamental group of a geometrically interesting closed manifold is very likely to be residually finite. However, there is an infinite

L -invariants of regular coverings of compact manifolds and CW-complexes

759

group r with four generators and four relations which has no finite quotient except the trivial one and hence satisfies F^-^ = {1} [119]. Meanwhile Theorem 5.1 has been generalized and put into somewhat different context in [56,74,92,229]. 6. L^-Betti numbers and groups In this section we explain some applications of L^-Betti numbers to group theory. Given a group r , we define its L^-Betti number bf\r) by bf\Er) where EF -^ BF is the universal principal F-bundle. This is only well-defined if the (p + l)-skeleton of BF is finite. The definition for arbitrary groups will be given in Definition 10.9. The next result of Cheeger and Gromov was proven in special cases by Gottlieb [104] andRosset[219]. 6.1 (Cheeger and Gromov [54, Corollary 0.6, p. 193]). Let F be a group such that there exists a subgroup F' of finite index whose classifying space BF' is a finite CWcomplex. Let Xvirt(^) be the rational-valued virtual Euler characteristic of the group F in the sense of [35, IX.7], [246]. Suppose that F contains an infinite normal amenable subgroup. Then THEOREM

Xvirt(n=0. By definition Xviit(^) = frTn * ^ (^^)mula of Theorem 1.7.2 PROOF.

We derive from the Euler-Poincare for-

Xv^rtCn = — i - . J2(-^y • bf{BF\ Now the claim follows from Theorem 4.1.

D

We mention the following result THEOREM 6.2 (Reich [213, Corollary 9.3]). Let F be an infinite group which belongs to LinnelVs class C introduced in Section 2 and has an upper bound on the orders of its finite subgroups. Suppose that there exists a subgroup F' of finite index whose classifying space BF' is a finite CW-complex. Then

Xvirt(r) ^ 0. Next we mention the following observation about Thompson's group F. It consists of orientation preserving dyadic PL-automorphisms of [0,1], where dyadic means that all slopes are integral powers of 2 and the break points are contained in Z[l/2]. It has the presentation F = (xo,x\,X2, " -{x- ^XfjXi =Xn-i-{ for / < n).

760

W. Luck

This group has some very interesting properties. It is not elementary amenable and does not contain a subgroup which is free on two generators [33,45]. Hence it is a very interesting question whether F is amenable. Since 5 F is of finite type [36], the L^-Betti numbers (2)

bp (F) are defined for all p^O. We conclude from Theorem 4.1 of Cheeger and Gromov (2)

that a necessary condition for F to be amenable is that bp (F) vanishes for all p^O. This motivates the following result. THEOREM 6.3 (Liick [157, Theorem 0.8]). All the L^-Betti numbers bf\F) of Thompson 's group F vanish. For the proof of Theorem 6.3 we need the next result. Given a selfmap / : F -> F, its mapping cylinder Mf is obtained by gluing the bottom of the cylinder F x [0,1] to F by the identification (jc, 0) = / ( x ) . Its mapping torus Tf is obtained from the mapping cylinder by identifying the top and the bottom by the identity. If / is a homotopy equivalence, Tf is homotopy equivalent to the total space of a fibration over S^ with fiber F. Conversely, the total space of such a fibration is homotopy equivalent to the mapping torus of the self homotopy equivalence of F given by the fiber transport with a generator of TT 1(5*). The homotopy type of Tf depends only on the homotopy class of / . There is an obvious map from Tf to S^ which induces an epimorphism ll'.7T\{Tf) -^ Z. THEOREM 6.4 (Luck [155, Theorem 2.1, p. 207]). Let F be a connected CW-complex of finite type and f: F ^ F be a selfmap. Let

/x:7ri(F/)^F^Z be a factorization of /JL into epimorphisms. Let Tf -^ Tf be the regular covering of Tf with r as group of deck transformations which is associated to (p. Then

bf{Tf)=0

forp^O.

PROOF. For simplicity we give here only the proof in the case where / is a homotopy equivalence and 0 is the identity on 7T\ (Tf), i.e., Tf is the universal covering Tf. Consider

any positive integer d. Let Fd be the preimage of dZ under /x:7T\(Tf) -> Z. Let Tf be the covering Tf -^ ^d\Tf with Fj as group of deck transformations. We get from the multiplicative property of Theorem 1.7.7

One easily checks that Fd\Tf is homotopy equivalent to Tfd. On Tfd there is a CWcomplex structure whose number of p-cells is bounded by the number C which is the sum of the numbers of /7-cells and the number of (p — l)-cells in F. Notice that C is

L -invariants of regular coverings of compact manifolds and CW-complexes

761

independent of d. Since the pih chain module in the cellular L^-chain complex of Tj-d is 0 , ^ 1 l^iPd), we conclude

bf\fj:)

= dimu^rd)K\W'))

^ dimAr(r.)(cf ( ^ ) ) = C.

Hence we have shown that for allJ ^ 1

Taking the limit for J ^^ oo finishes the proof of Theorem 6.4.

D

Next we give the proof of Theorem 6.3. There is a subgroup F\ c F together with a monomorphism 0:F\-^ F\ such that F\ is isomorphic to F and F is the HNN-extension of F\ with respect to